Martindale Quotients of Jordan Algebras

José A. Anquela ^{1}^{, 2}

                    anque@orion.ciencias.uniovi.es

Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain Fax number: ++34 985 102 886

Kevin McCrimmon ^{2}

                          kmm4m@virginia.edu

Department of Mathematics, University of Virginia Charlottesville, VA 22904-4137, U.S.A.

Abstract: In this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter.

Keywords: Jordan algebra, Martindale quotient, denominator filter of ideals.

MSC2000: 17C10.

Introduction

In the last years, several authors have studied algebras of quotients of Jordan systems. The origin can be set in [8], where Martínez gives necessary and sufficient conditions for a Jordan algebra to have an algebra of fractions. She uses the Tits-Kantor-Koecher construction to move the problem into the Lie algebra setting, so that only rings of scalars containing 1/6 can be considered. In a similar fashion, taking into account Siles’ work on quotients of Lie algebras [15], García and Gómez-Lozano [4] give a notion of Martindale-like quotient for linear Jordan systems over fields with respect to filters of ideals and prove the existence of maximum quotients in the nondegenerate case. In [3], the restriction on the rings of scalars is weakened to having 1/2, though only strongly prime algebras are considered. However, a full description of the maximum Martindale-like quotients of strongly prime Jordan algebras is obtained, giving a new unified approach to Zelmanovs’s classification theorems [16]. In [14], Montaner gives a Jordan version of Lambek and Utumi’s algebras of quotients over arbitrary rings of scalars, but only for nondegenerate algebras. His notion includes that of García and Gómez-Lozano in the case of algebras.

1 Corresponding author. Partly supported by the Plan de Investigación del Principado de Asturias, FICYT IB05-017.

2 Partly supported by the Ministerio de Educación y Ciencia and Fondos FEDER, MTM2004-06580-C02-01 and MTM2007-62390.

In this paper we give a (quadratic) notion of Martindale quotient Q for arbitrary Jordan algebras J with respect to denominator filters of ideals. Unlike more general theories of localization in the associative case, we demand that a Martindale quotient algebra Q contains a faithful copy of the original algebra J. We impose no conditions (such as semiprimeness or nondegeneracy), only that the “denominators” are faithful to J (sturdy). This notion extends that given in the linear setting by García and Gómez-Lozano [3], and also includes the notion of Martindale-like cover [1, 2] for nondegenerate algebras. Since we do not assume any regularity condition other than the existence of a denominator filter of ideals, we cannot make use of the structure theory of nondegenerate Jordan algebras, unlike [1, 2, 3, 14]. We generalize all basic properties known in the linear case [3, 4] and even show that each Martindale quotient of a denominatored algebra is contained in a maximal one, though we leave open the problem of uniqueness of those maximal quotients.

The paper is divided into six sections, apart from a preliminary one recalling basic results and terminology. The way elements of Martindale quotients are boosted into the original algebra by denominator ideals is deeply related to annihilators, so we start in the first section with some combinatorial results concerning annihilation by powers of ideals. The second section defines Martindale quotients with respect to denominator filters of ideals and studies basic properties leading to the notion of maximum Martindale quotient. In the third section we give several examples, including a degenerate example, and prove that our notion is the quadratic generalization of that given in [4] and includes that of Martindale-like cover [1, 2]. In Section 4, we exhibit a way to build Martindale quotients out of any extension of Jordan algebras. This construction is extensively used in the next section where we show that there is a bound on the cardinality of Martindale quotients of a given denominatored algebra, implying the existence of maximal quotients; existence of a maximum quotient is equivalent to directedness of the lattice of quotients. Finally, the last section deals with the interaction between Martindale quotients and unital hulls.

0. Preliminaries

0.1 We will deal with Jordan algebras over a ring of scalars Φ. The reader is referred to [5, 7, 12] for definitions and basic properties not explicitly mentioned or proved in this section. Given a Jordan algebra J, its products will be denoted x^{2}, U_{x}y, for x, y ∈ J. They are quadratic in x and linear in y and have linearizations denoted V_{x}y = x∘y, U_{x,z}y = {x, y, z} = V_{x,y}z, respectively. For y ∈ J, the quadratic operator ∩_{y} : J ⟶ J of inner multiplication by y is given by ∩_{y}(x) = U_{x}y. Each Jordan algebra is imbedded in its free unital hull Ĵ:= Φ1 ⊕ J. Zelmanov’s structure theory shows that the proper unital hulls are those which are tight; in (6.2) we will tighten Ĵ to get the “true” unital hull J̌.

0.2 We recall the following identities valid for arbitrary Jordan algebras which will be needed in the sequel:

(i) x^{2} ∘ z = {x, x, z},

(x ∘ y) ∘ z = {x, y, z} + {y, x, z},

(ii) {U_{b}a, a, y} = {b, U_{a}b, y},

{x, U_{a}b, y} = {{x, a, b}, a, y} − {b, U_{a}x, y},

{U_{b}a, x, y} = {b, {a, b, x}, y} − {U_{b}x, a, y},

(iii) U_{x}U_{y}z +U_{y}U_{x}z −U_{x∘y}z = −V_{x,y}V_{y,x}z +(U_{x}y^{2})∘z = {U_{x}y, z, y}−V_{x}U_{y}V_{x}z,

U_{y}x^{2} = (x ∘ y)^{2} − U_{x}y^{2} − y ∘ U_{x}y = (x ∘ y)^{2} + U_{x}y^{2} − {x, y, x ∘ y},

(iv) U_{U_{x}y} = U_{x}U_{y}U_{x},  (U_{x}y)^{2} = U_{x}U_{y}x^{2},  U_{x^{2}} = U_{x}U_{x},

(v) (U_{x}y) ∘ z = {x, y, x ∘ z} − U_{x}(y ∘ z),

(vi) 2U_{x}z = (x ∘ z) ∘ x − x^{2} ∘ z,

(vii) [V_{x,y}, V_{z,w}] = V_{{x,y,z},w} − V_{z,{y,x,w}},

(viii) U_{x}U_{a,b} = V_{x,b}V_{x,a} − V_{U_{x}b,a},

(ix) U_{{x,y,z}} + U_{U_{x}y,U_{z}y} = U_{x}U_{y}U_{z} + U_{z}U_{y}U_{x} + U_{x,z}U_{y}U_{x,z}.

Indeed, (i), (iii–vi), and the first part of (ii) follow from Macdonald’s Theorem [6], the second and third identities of (ii) follow from the first one by linearization, and (vii), (viii), (ix) are respectively JP15, JP13, JP20 in [7].

0.3 A Jordan algebra J is said to be nondegenerate if zero is the only absolute zero divisor, i.e., the only x ∈ J such that U_{x} = 0.

0.4 We recall that an inner ideal I of a Jordan algebra J is a Φ-submodule of J satisfying U_{I} Ĵ ⊆ I [i.e., U_{I}J + I^{2} ⊆ I], while an outer ideal of J is a Φ-submodule I of J satisfying U_{J}I ⊆ I [i.e., U_{J}I + I ∘ J ⊆ I], which implies {I, J, J} ⊆ I by (0.2)(i). We say that I is an ideal of J if it is both an inner and outer ideal. The cube I^{3} = U_{I}I and the product U_{I}L of ideals I, L of J are again ideals of J [10, p. 221].

0.5 Given elements x, y in a Jordan algebra J, the symmetric sets of three expressions

β_{x}(y) := {U_{x}y, ∩_{x}y, V_{x}y} = {∩_{y}x, U_{y}x, V_{y}x} =: β_{y}(x)

(the three basic Jordan products of x and y) will appear frequently. For any subsets S, T, L we will call the set

β_{S}(T) := ⋃_{x∈S,y∈T} β_{x}(y)

the basic S-boost of T, and

\begin{matrix}Z_{L;S}(T) := & β_{S}(T) ∪ \{\{x, y, z\}| x ∈ S, y ∈ T, z ∈ L\}∪ \\ \{U_{x}U_{y}z| x ∈ S, y ∈ T, z ∈ L\} ∪ \{U_{x}y^{2}| x ∈ S, y ∈ T \} \end{matrix}

the Zelmanov S-boost of T in L. When any of the subsets S, T, L above consists of a single element x, we will write x instead of {x}. In the same fashion, “≠ 0”, “= 0” will be abbreviations of “≠ {0}”, “= {0}”, respectively.

0.6 We say that a Jordan algebra J is semiprime if I^{3} ≠ 0, for any nonzero ideal I of J, and say that J is prime if U_{I}L ≠ 0, for any nonzero ideals I, L of J. Every nondegenerate Jordan algebra is semiprime. An ideal I of J is said to be essential if I ∩ L ≠ 0 for any nonzero ideal L of J. It is obvious that the intersection of two essential ideals of J is again an essential ideal. Moreover, if I, K are essential ideals of a semiprime J, the product U_{I}K is essential: for any nonzero ideal L of J, L ∩ I ∩ K ≠ 0, hence 0 ≠ (L ∩ I ∩ K)^{3} ⊆ L ∩ U_{I}K.

0.7 In a Jordan algebra J, the Zelmanov annihilator Zann_{J}(T) of a subset T of J is the set of all z ∈ J such that Z_{J;z}(T) = Z_{J;T} (z) = 0, i.e., for all x ∈ T, (Z1) U_{z}x = 0, (Z2) U_{x}z = 0, (Z3) V_{z,x}Ĵ = 0, (Z3)′ V_{x,z}Ĵ = 0, (Z4) U_{z}U_{x}Ĵ = 0, (Z4)′ U_{x}U_{z}Ĵ = 0. Here (Z3) ⇔ (Z3)′ by (0.2)(i), and in its presence (Z4) ⇔ (Z4)′ by (0.2)(iii), so Z_{J;z}(x) = 0 ⟺ Z_{J;x}(z) = 0. Avoiding the unital hull, Zann_{J}(x) is the set of z which satisfy (Z1), (Z2), (Z3a) z ∘ x = 0, (Z3b) {z, x, J} = 0, (Z4a) U_{z}U_{x}J = 0, (Z4b) U_{z}x^{2} = 0. Thus

\begin{aligned}Zann_{J}(T) & = \{z ∈ J | U_{z}T = U_{T} z = \{z, T, Ĵ\} = U_{z}U_{T} Ĵ = 0\} \\ & = \{z ∈ J | Z_{J;z}(T) = 0\} = \{z ∈ J | Z_{J;T} (z) = 0\} \end{aligned}

(if 1/2 ∈ Φ then {z, T, Ĵ} = 0 suffices [9, 1.4]). This is always an inner ideal, and is an ideal if T is an ideal of J [9, 1.4 p. 235]. We say T is sturdy if Zann_{J}(T) = 0. When T = I is an ideal, the condition U_{z}U_{I} Ĵ = 0 follows from U_{z}I = 0. If I ∩Zann_{J}(I) = 0 then Zann_{J}(I) is the maximum ideal of J missing I: if I ∩ K = 0 for an ideal K of J, then Z_{J;I}(K) ⊆ I ∩ K = 0 ⟹ K ⊆ Zann_{J}(I). This implies that sturdy ideals are always essential. Since for any ideal I of J the ideal L := I ∩ Zann_{J}(I) has L^{3} = 0, we have

(1) essential ideals coincide with sturdy ideals in semiprime Jordan algebras.

When I is an ideal of a nondegenerate J,

(2) Zann_{J}(I) = {z ∈ J | U_{z}I = 0} = {z ∈ J | U_{I}z = 0} (J nondegenerate)

(see [10, 1.2a, 1.7; 13, 1.3]).

1. Technical Lemmas Concerning the Annihilator

1.1 Lemma. Let Q be a Jordan algebra, I and S submodules of Q, and set S′ = S ∘ I + S.

(i) If q ∈ Q has U_{I}q + q ∘ I ⊆ S then

(a) {q, I, I} ⊆ S′, (b) V_{q,I^{3}} + V_{I^{3},q} ⊆ V_{S′,I} + V_{I,S′},

and, if in addition U_{q}I + U_{q}I^{3} ⊆ S, then

(c) U_{q}U_{I^{3}} ⊆ U_{S′}U_{I} + U_{I}U_{S} + U_{S,I^{3}}U_{I} + U_{q,I}U_{S,I^{3}} + (V_{S′,I} + V_{I,S′})^{2} + V_{S,I^{3}}.

(ii) Annihilation of ideals and boosting them to 0 are closely related. Indeed, if I is a submodule of Q with I^{3} ⊆ I, then

q ∈ Zann_{Q}(I) ⟹ β_{q}(I) = 0 (equivalently β_{I}(q) = 0) ⟹ q ∈ Zann_{Q}(I^{3}).

(iii) If I ⊆ J ⊆ Q where I is an ideal in the subalgebra J and q ∈ Q satisfies β_{q}(I) ⊆ J (equivalently β_{I}(q) ⊆ J), then V_{q,(I^{3})^{3}} + V_{(I^{3})^{3},q} ⊆ V_{I,I}.

Proof: (i) By (0.2)(i) {q, I, I} ⊆ (q ∘ I) ∘ I + U_{I,I}q ⊆ S ∘ I + S = S′ as in (a), so by (0.2)(ii) for a, b ∈ I and x = q, V_{q,U_{a}b} = V_{{q,a,b},a} − V_{b,U_{a}q} ∈ V_{S′,I} + V_{I,S} and dually V_{U_{b}a,q} = V_{b,{a,b,q}} − V_{U_{b}q,a} ∈ V_{I,S′} + V_{S,I} as in (b). For (c), U_{I^{3}} is spanned by all U_{c}, U_{c,c′} for c = U_{a}b, c′ = U_{a′}b′ ∈ I^{3} for a, a′, b, b′ ∈ I, and we have U_{q}U_{c} = U_{q}U_{a}U_{b}U_{a} [by (0.2)(iv)] = (U_{{q,a,b}} + U_{U_{q}a,U_{b}a} − U_{b}U_{a}U_{q} − U_{q,b}U_{a}U_{q,b})U_{a} [by (0.2)(ix)] ⊆ U_{S′}U_{I} + U_{S,I^{3}}U_{I} + U_{I}U_{S} + U_{q,I}U_{S,I^{3}} [by respectively (a); U_{q}a ∈ S; (0.2)(iv) for y = q and U_{a}q ∈ S; y → q, b in linearized (0.2)(iv) and U_{a}q ∈ S], while U_{q}U_{c,c′} = V_{q,c′}V_{q,c} −V_{U_{q}c′,c} [by (0.2)(viii) for x = q, a = c, b = c′] ⊆ (V_{S′,I} +V_{I,S′})^{2} + V_{S,I^{3}} [by (b) and hypothesis U_{q}I^{3} ⊆ S].

(ii) The first implication is obvious. For the second, β_{q}(I) = 0 ⟹ β_{q}(I^{3}) = 0 and {q, I^{3}, Q̂} = V_{q,I^{3}}(Q̂) = 0 and U_{q}U_{I^{3}} Q̂ = 0 by applying (i)(b), (i)(c) to Q̂ with S = 0.

(iii) V_{q,(I^{3})^{3}} + V_{(I^{3})^{3},q} ⊆ V_{J,I^{3}} + V_{I^{3},J} [by (i)(b) for I^{3} in place of I and S = J] ⊆ V_{I,I} [by (i)(b), for S = I and all q ∈ J, because β_{q}(I) ⊆ I since I is an ideal in J].

1.2 Lemma. Let Q be a Jordan algebra, J a subalgebra of Q, q ∈ Q, and I an ideal of J with β_{I}(q) = 0. Then β_{I^{3}}(β_{J}(q)) = 0 so β_{J}(q) ⊆ Zann_{Q}((I^{3})^{3}).

Proof: Let L := I^{3}, x ∈ J. Note V_{L,q}Q̂ = V_{q,L}Q̂ = 0 by (1.1)(i)(b) with S = 0, and {q, J, L} = q ∘ (J ∘ L) − {q, L, J} [by (0.2)(i)] ⊆ V_{q,L}1 − V_{q,L}J [since J ∘ L ⊆ L by idealness of L in J] = 0. Now we check V_{z,L}Q̂ = 0 successively for z = x ∘ q, U_{x}q, U_{q}x in β_{J}(q):

V_{x∘q,L} = [V_{x}, V_{q,L}] + V_{q,x∘L} [by (0.2)(vii) with y = 1] = 0 since x ∘ L ⊆ L,
V_{U_{x}q,L} = −V_{U_{x}L,q} + V_{x,{q,x,L}} [by (0.2)(ii)] ⊆ V_{L,q} + V_{x,{q,J,L}} = 0,
V_{U_{q}x,L} = −V_{U_{q}L,x} + V_{q,{x,q,L}} [by (0.2)(ii))] ⊆ V_{β_{I}(q),J} + V_{Q,V_{L,q}J} = 0.

Now we have β_{J}(q) ∘ L = V_{β_{J} (q),L}ˆ1 = 0 and it remains to show U_{L}β_{J}(q) = U_{β_{J} (q)}L = 0. For all k ∈ L we have by (0.2)(iii), (iii), (v) respectively that

U_{k}(U_{x}q) = −U_{x}U_{k}q + U_{x∘k}q − {{x, k, q}, x, k} + q ∘ (U_{k}x^{2})
⊆ U_{J}U_{I}q + U_{I}q + {V_{q,L}(J), J, J} + q ∘ I = 0,
U_{k}(U_{q}x) = −U_{q}U_{k}x + U_{k∘q}x − {{k, q, x}, k, q} + x ∘ (U_{q}k^{2})
⊆ U_{q}I + U_{I∘q}J + V_{q,L}Q + J ∘ (U_{q}I) = 0,
U_{k}(x ∘ q) = −x ∘ (U_{k}q) + {k, q, k ∘ x} ⊆ J ∘ U_{I}q + U_{I}q = 0,

and similarly we have by (0.2)(iv), (iv), (iii) respectively that

U_{U_{x}q}k = U_{x}U_{q}U_{x}k ⊆ U_{J}U_{q}I = 0,
U_{U_{q}x}k = U_{q}U_{x}U_{q}k ⊆ U_{q}U_{J}U_{q}I = 0,
U_{x∘q}k = U_{x}U_{q}k + U_{q}U_{x}k + {{x, q, k}, x, q} − k ∘ (U_{q}x^{2})
⊆ U_{J}U_{q}I + U_{q}I + {V_{L,q}J, J, Q} + L ∘ U_{q}J = 0.

We have proved β_{I^{3}}(β_{J}(q)) = 0, so β_{J}(q) ⊆ Zann_{Q}((I^{3})^{3}) follows from (1.1)(ii).

2. Denominatored Algebras and Martindale Quotients

2.1 Given a Jordan algebra J, a nonempty set F of ideals of J will be called a filter if, for any K, L ∈ F, there exists I ∈ F such that I ⊆ U_{K}L (so that I ⊆ K ∩L). Notice that, in particular, for any K ∈ F, there exists K′ ∈ F such that K′ ⊆ K^{3}. We say that a filter F′ is finer than F (F′ ≻ F) if for all I ∈ F there exists I′ ∈ F′ with I′ ⊆ I (for example, if F′ ⊇ F), and F, F′ are cofinal if F′ ≻ F ≻ F′. For a filter F, its closure \overline{F}, consisting of all ideals of J which contain some ideal of F, is a filter which contains F and, moreover, F and \overline{F} are cofinal. Notice that

F′ ≻ F ⟺ \overline{F′} ⊇ \overline{F}.  (1)

A filtered algebra will be a pair (J, F) where J is a Jordan algebra and F is a filter of ideals of J. If a filter F consists of sturdy ideals of J, then it will be called a denominator filter, and the pair (J, F) will be called a denominatored algebra. Notice that the closure of a denominator filter is also a denominator filter.

For example, the set of essential ideals of a semiprime Jordan algebra is a denominator filter by (0.6) and (0.7)(1).

We also remark that, for linear Jordan algebras (1/2 ∈ Φ), a nonempty set of ideals of J is a denominator filter if and only if it is a power filter of sturdy ideals in the sense of [3, 1.2]

2.2 A Martindale quotient of a denominatored algebra (J, F) is a pair (Q, τ) where Q is a Jordan algebra, and τ : J ⟶ Q is an algebra monomorphism such that for any 0 ≠ q ∈ Q there exists I ∈ F which boosts q nontrivially into J, in the sense that

0 ≠ β_{τ(I)}(q) ⊆ τ(J),

which implies {τ(I), τ(I), q} ⊆ τ(J) by (0.2)(i).

2.3 Remark: Notice that (1) any Martindale quotient (Q, τ) of a denominatored algebra (J, F) is tight over τ(J) (or, by abuse of language, tight over J) in the sense that every nonzero ideal of Q hits τ(J): if 0 ≠ L is an ideal of Q, we can take any 0 ≠ q ∈ L, and there exists I ∈ F such that 0 ≠ β_{τ(I)}(q) ⊆ τ(J) ∩ L.

Also, (2) The denominator filter F in J induces a denominator filter F̃ = {Ĩ| Ĩ is an ideal of Q, Ĩ ⊇ τ(I) for some I ∈ F} on Q, since all such Ĩ are sturdy in Q: if Ĩ ⊇ τ(I) then 0 = τ(Zann_{J}(I)) [by sturdiness of I] = Zann_{τ(J)}(τ(I)) [since τ is an algebra isomorphism of J with τ(J)] ⊇ τ(J) ∩ Zann_{Q}(Ĩ) forces Zann_{Q}(Ĩ) (which is an ideal of Q) to vanish by (1).

2.4 Remark: Any covering map of Martindale quotients must be injective: if (Q, τ) is a Martindale quotient of a denominatored algebra (J, F) and τ′ : J ⟶ Q′ a Jordan algebra monomorphism, then any algebra homomorphism f : Q ⟶ Q′ which satisfies fτ = τ′ must be injective; in particular, this holds when (Q′, τ′) is another Martindale quotient of (J, F) and fτ = τ′. Indeed, Ker τ′ = 0 implies Ker f ∩ τ(J) = 0, hence Ker f = 0 by tightness (2.3)(1).

2.5 Proposition. Let (J, F) be a filtered algebra. Then, (J, F) is a denominatored algebra if and only if β_{I}(x) ≠ 0 for all 0 ≠ x ∈ J and I ∈ F. As a consequence, if (J, F) is a denominatored algebra, then (J, Id_{J}) is a Martindale quotient of (J, F).

Proof: Assume that F is a denominator filter. If x ∈ J satisfies β_{I}(x) = 0, for some I ∈ F, then x ∈ Zann_{J}(I^{3}) by (1.1)(ii). On the other hand, there is I′ ∈ F such that I′ ⊆ I^{3}. But this means x ∈ Zann_{J}(I′), which forces x = 0 since I′ is sturdy. The converse is clear and the consequence is straightforward.

2.6 Proposition. Let (Q, τ) be a Martindale quotient of a denominatored algebra (J, F).

(i) All F-boosts are nontrivial: β_{τ(I)}(q) ≠ 0 for all 0 ≠ q ∈ Q and I ∈ F.

(ii) In particular, Zann_{Q}(τ(I)) = 0 and τ(I) remains sturdy in Q, for all I in F.

(iii) Cofinal denominator filters have precisely the same Martindale quotients. Moreover, (Q, τ) remains a Martindale quotient of (J, F′) for any denominator filter F′ ≻ F.

Proof: (i) Suppose on the contrary that 0 ≠ q ∈ Q satisfies β_{τ(I)}(q) = 0 for some I ∈ F, hence by (1.2) applied to τ(J) ⊆ Q we have β_{τ(J)}(q) ⊆ Zann_{Q}((τ(I)^{3})^{3}). There exists K ∈ F satisfying 0 ≠ β_{τ(K)}(q) ⊆ τ(J). The filter F contains I′ ⊆ U_{I}I = I^{3} and I′′ ⊆ U_{I′}I′ ⊆ (I^{3})^{3}, so τ(I′′) ⊆ (τ(I)^{3})^{3} and 0 ≠ β_{τ(K)}(q) ⊆ τ(J) ∩ β_{τ(J)}(q) ⊆ τ(J) ∩ Zann_{Q}(τ(I′′)) ⊆ Zann_{τ(J)}(τ(I′′)). But since τ is an injective algebra homomorphism, Zann_{τ(J)}(τ(I′′)) = τ(Zann_{J}(I′′)) = 0 by sturdiness of I′′, which is a contradiction.

(ii) is a direct consequence of (i).

(iii) The fact that cofinal denominator filters have the same Martindale quotients readily follows from (i). It is clear that if F_{1} ⊆ F_{2} are denominator filters in J, all Martindale quotients of (J, F_{1}) are also Martindale quotients of (J, F_{2}). Since F ⊆ \overline{F}, (Q, τ) is a Martindale quotient of (J, \overline{F}), hence of (J, \overline{F′}) because \overline{F′} ⊇ \overline{F} by (2.1)(1). Thus (Q, τ) is a Martindale quotient of (J, F′) since F′ and \overline{F′} are cofinal.

Notice that (iii) shows (J, F) and its closure (J, \overline{F}) have precisely the same Martindale quotients. Since always U_{I}K, I ∩ K ∈ \overline{F} for I, K ∈ \overline{F}, this shows that without loss of generality we could have required a denominator filter to be a filter in the set-theoretic sense (closed under enlargements I′ ⊇ I and intersections I ∩ K), which is also closed under products U_{I}K.

2.7 Proposition. Let (Q′, τ′) be a Martindale quotient of a denominatored algebra (J, F), and let τ : J ⟶ Q, f, g : Q ⟶ Q′ be algebra homomorphisms with fτ = gτ = τ′. Then f and g agree on any q ∈ Q boostable into τ(J):

β_{τ(I)}(q) ⊆ τ(J) for some I ∈ F ⟹ f(q) = g(q).

In particular, if Q, Q′ are both Martindale quotients then any f : Q ⟶ Q′ with fτ = τ′ is unique and injective, so that if (Q, τ) = (Q′, τ′) then f = Id_{Q}.

Proof: Our goal is to prove that q′ = f(q) − g(q) ∈ Q′ vanishes. By (2.6)(i) it will suffice to prove

β_{τ′(L)}(q′) = 0 for L ∈ F, L ⊆ I^{3}.

As usual, we show q′ is killed by all three pieces of β_{τ′(L)}. Using fτ = gτ = τ′ on J, for any k ∈ I we have

U_{τ′(k)}q′ = U_{τ′(k)}f(q) − U_{τ′(k)}g(q) = U_{fτ(k)}f(q) − U_{gτ(k)}g(q)
= f(U_{τ(k)}q) − g(U_{τ(k)}q) ∈ (f − g)(τ(J)) = 0
τ′(k) ∘ q′ = τ′(k) ∘ f(q) − τ′(k) ∘ g(q) = fτ(k) ∘ f(q) − gτ(k) ∘ g(q)
= f(τ(k) ∘ q) − g(τ(k) ∘ q) ∈ (f − g)(τ(J)) = 0,

and for any k ∈ L we have

U_{q′}τ′(k) = U_{f(q)−g(q)}τ′(k) = (−U_{f(q)} + U_{g(q)} + U_{f(q),f(q)−g(q)})τ′(k)
= −U_{f(q)}fτ(k) + U_{g(q)}gτ(k) − {f(q), τ′(k), q′}
= −f(U_{q}τ(k)) + g(U_{q}τ(k)) − {q′, τ′(k), f(q)}
∈ (g − f)(τ(J)) − {q′, τ′(L), Q′} = 0

[using the above to replace S, I, Q, q in (1.1)(i)(b) by 0, τ′(I), Q′, q′, respectively, noticing τ′(L) ⊆ τ′(I^{3}) = (τ′(I))^{3}].

When Q, Q′ are Martindale quotients, then all q are boostable, so f = g is unique, and it is injective by (2.4).

2.8 A Martindale quotient (Q, τ) of a denominatored algebra (J, F) will be called a maximum if for any other Martindale quotient (Q′, τ′) of (J, F), there exists an algebra homomorphism f : Q′ ⟶ Q such that fτ′ = τ.

The following result is a consequence of (2.7).

2.9 Theorem (Universal Property for Maximum Martindale Quotients). Let (Q, τ) be a maximum Martindale quotient of a denominatored algebra (J, F). If (Q′, τ′) is a Martindale quotient of (J, F), then there exists a unique algebra homomorphism f : Q′ ⟶ Q such that fτ′ = τ. Moreover, f is necessarily injective. Thus maximum Martindale quotients of a given denominatored algebra (J, F) are unique up to isomorphism.

Proof: If (Q, τ), (Q′, τ′) are both maximum Martindale quotients of (J, F), then the unique algebra homomorphisms f : Q′ ⟶ Q, f′ : Q ⟶ Q′, such that fτ′ = τ, f′τ = τ′ are mutually inverse isomorphisms since by (2.7) f ∘ f′ = Id_{Q} and f′ ∘ f = Id_{Q′}.

3. Examples

3.1 Subexample. If (Q, τ) is a Martindale quotient of a denominatored algebra (J, F) then so is (Q′, τ′) for every subalgebra Q′ of Q with τ(J) ⊆ Q′, where τ′ denotes the restriction of τ.

This shows that in general there will be lots of “smaller” quotients (think of rational numbers with denominators restricted to a multiplicatively closed subset of the subset of the integers). The more interesting question is whether there are larger Martindale quotients (see Section 5).

3.2 Sturdy Ideal Example. If (J, F) is a denominatored Jordan algebra where J is an ideal of Q and all I ∈ F remain sturdy in Q (Zann_{Q}(I) = 0 for all I ∈ F), then (Q, τ), for τ the inclusion map, is a Martindale quotient of (J, F).

Indeed, always β_{I}(q) ⊆ J, and by (1.1)(ii) β_{I}(q) = 0 ⟹ q ∈ Zann_{Q}(I^{3}) = 0 because I^{3} contains an ideal L of F which remains sturdy in Q.

3.3 Lemma. If (J, F) is a unital denominatored Jordan algebra, then any Martindale quotient (Q, τ) of (J, F) is unital with the same unit as J: 1_{Q} = τ(1_{J}).

Proof: We shall use Peirce decompositions (see [7, Section I.5]). Indeed e = τ(1_{J}) is an idempotent of Q such that τ(J) ⊆ Q_{2}(e), and for any q ∈ Q_{1}(e) ∪ Q_{0}(e), β_{τ(J)}(q) ⊆ U_{q}τ(J) + U_{τ(J)}q + q ∘ τ(J) ⊆ U_{q}Q_{2}(e) + U_{Q_{2}(e)}q + q ∘ Q_{2}(e) ⊆ Q_{0}(e) + 0 + Q_{1}(e). Thus, for any I ∈ F such that β_{τ(I)}(q) ⊆ τ(J), we have that β_{τ(I)}(q) ⊆ Q_{2}(e) ∩ (Q_{0}(e) + Q_{1}(e)) = 0, which implies q = 0 by (2.6)(i). This shows that Q = Q_{2}(e), i.e., e is the unit element of Q.

3.4 Simple Examples. If J is a unital Jordan algebra and F = {J} (when J is simple this is the only possible filter), then the only Martindale quotient is, up to isomorphism, (Q, τ) = (J, Id). If J is simple but not necessarily unital, F = {J} is the unique denominator filter of J, but now there can be many quotients: if J = A^{(+)} for A ⊆ End(V_{Δ}) the ideal of finite-rank endomorphisms of an infinite-dimensional right vector space over a division ring Δ, then any Jordan subalgebra Q with A^{(+)} ⊆ Q ⊆ End(V_{Δ})^{(+)} is a Martindale quotient of (J, F).

For a unital J, F = {J}, and (Q, τ) a Martindale quotient of (J, F), we have by (3.3) q = U_{τ(1)}q ∈ U_{τ(J)}q ⊆ τ(J), so Q = τ(J).

When J = A^{(+)}, as above, since J is an ideal of Q, we can use (3.2) as soon as we check that the lone ideal I = J in F is sturdy in Q: if q ∈ Q is nonzero, then there exists v ∈ V such that q(v) = w ≠ 0 and U_{q}a(v) = qaq(v) = w for any finite rank transformation a with a(w) = v, so U_{q}a ≠ 0 and q ∉ Zann_{Q}(J).

3.5 Nondegenerate Examples. Let J be a nondegenerate Jordan algebra and F be the set of all essential ideals of J (which is a denominator filter of J (2.1)), and let τ : J ⟶ Q be a Jordan algebra monomorphism. Then (Q, τ) is a Martindale quotient of (J, F) iff Q is a Martindale-like cover of τ(J) in the sense of [2, 2.1, 2.4] (i.e. for any 0 ≠ q ∈ Q there exists an essential I such that β_{τ(I)}(q) ⊆ τ(J) and U_{τ(I)}q ≠ 0).

Sufficiency is obvious and necessity follows from the following general observation (improving on (2.6)(i)).

3.6 Lemma. If (Q, τ) is a Martindale quotient for a nondegenerate denominatored Jordan algebra (J, F) then U_{τ(I)}q ≠ 0 for all 0 ≠ q ∈ Q and I ∈ F.

Proof: Replacing J by τ(J), we may assume J ⊆ Q and τ is the inclusion map. Since Q is tight over J (2.3), Q is also nondegenerate [11, 2.9(iii)]. Given 0 ≠ q ∈ Q and I ∈ F there is L ∈ F satisfying β_{L}(q) ⊆ J, and by the definition of filter we may choose such an L with L ⊆ I. We claim that U_{L}q ≠ 0 (hence U_{I}q ≠ 0 too). Otherwise, U_{L}q = 0, so U_{L}U_{q}L = 0 by nondegeneracy and [1, 3.4], which implies U_{q}L ⊆ Zann_{J}(L) by (0.7)(2), but Zann_{J}(L) = 0 by sturdiness of L, hence U_{q}L = 0. Also U_{L}(L ∘ q) ⊆ L ∘ U_{L}q + {L ∘ L, q, L} ⊆ L ∘ U_{L}q + U_{L}q = 0 by (0.2)(v) implies L∘q ⊆ Zann_{J}(L) = 0 by (0.7)(2) again. Thus β_{L}(q) = 0, which contradicts (2.6)(i).

3.7 Linear Examples. For a denominatored linear Jordan algebra (J, F) (1/2 ∈ Φ) and a Jordan algebra monomorphism τ : J ⟶ Q, (Q, τ) is a Martindale quotient of (J, F) iff it is an algebra of Martindale-like quotients of J with respect to F in the sense of [3, 1.3].

Proof: As above, we may assume J ⊆ Q. We must show that the Martindale quotient condition 0 ≠ β_{I}(q) ⊆ J and the Martindale-like quotient condition 0 ≠ I ∘ q ⊆ J for a given 0 ≠ q ∈ Q are equivalent. If Q is a Martindale quotient and 0 ≠ β_{I}(q) ⊆ J we claim q ∘ I ⊆ J is nonzero, since otherwise U_{I}q = 2U_{I}q ⊆ (I ∘ (I ∘ q) − I^{2} ∘ q) [by (0.2)(vi)] = 0, and any I′ ∈ F such that I′ ⊆ I^{3} would have β_{I′}(q) ⊆ U_{I}q + 2U_{q}I^{3} + q ∘ I ⊆ 0 + V_{q,I^{3}}q + 0 = 0 [by (1.1)(i)(b) with S = 0], which contradicts (2.6)(i).

Conversely, if Q is a Jordan algebra of Martindale-like quotients of J with respect to F, for a given 0 ≠ q ∈ Q, there are K, L ∈ F satisfying 0 ≠ K ∘ q ⊆ J, L ∘ q^{2} ⊆ J, and we claim 0 ≠ β_{I}(q) ⊆ J for any I ⊆ L ∩ K^{3} in F: 0 ≠ I ∘ q [by [3, 1.5]] ⊆ K ∘ q ⊆ J and by (0.2)(vi) again both U_{I}q ⊆ K ∘ (K ∘ q) − K^{2} ∘ q ⊆ J and U_{q}I ⊆ (q ∘ I) ∘ q − q^{2} ∘ I ⊆ (q ∘ U_{K}K) ∘ q − q^{2} ∘ L ⊆ K ∘ q − q^{2} ∘ L ⊆ J [since by (0.2)(v) q ∘ U_{K}K ⊆ {K, K, q ∘ K} − U_{K}(q ∘ K) ⊆ {K, K, J} − U_{K}J ⊆ K].

3.8 Admonitory Example. We give an example to show that in characteristic 2 there can be unexpectedly large “quotients” involving weird quadratic forms. Consider a Jordan algebra J = Φe ⊕ M for Φe ≅ Φ whose Peirce 1-space J_{1} = M relative to the idempotent e is a trivial bimodule, M^{2} = U_{M}M = 0. Then F = {J} is a denominator filter, and J imbeds naturally as an ideal in a unital special Martindale quotient algebra

E := J ⊕ E_{0} = Φe ⊕ M ⊕ E_{0} = ( \begin{matrix}Φ & 0 \\ M & E_{0} \end{matrix} )^{(+)} ↪ ( \begin{matrix}Φ & M^{∗} \\ M & E_{0} \end{matrix} )^{(+)} ≅ End_{Φ}(J)^{(+)}

for E_{0} := End_{Φ}(M) under

\begin{aligned}U_{αe⊕m⊕T_{0}}(βe ⊕ n ⊕ S_{0}) & := α^{2}βe ⊕ (αβm + αT_{0}(n) + T_{0}S_{0}(m)) ⊕ T_{0}S_{0}T_{0}, \\ (αe ⊕ m ⊕ T_{0})^{2} & := α^{2}e ⊕ (αm + T_{0}(m)) ⊕ T_{0}^{2}. \end{aligned}

But in the presence of 2-torsion there can be larger unnatural quotients. Denote by RS_{Φ}(M) := {λ ∈ Φ | λM = 0, λ^{2} = 2λ = 0}, the ideal in Φ of scalars in the radical of the squaring quadratic form λ → λ^{2} which kill M, and let WQ_{Φ}(M) := {weird quadratic maps ω : M → RS_{Φ}(M) | ω(M, M) = 0}. For convenience we will assume our quotients contain J and the imbedding τ is inclusion.

3.9 Proposition. (1) The Martindale quotients Q for (J, F) as above (J = Φe ⊕ M, M = J_{1}(e), M^{2} = U_{M}M = 0, F = {J}) are precisely all Q = J ⊕ Q_{0}, where Q_{0} is a Jordan algebra, with multiplication given by the Product Formula

\begin{aligned}U_{x⊕q_{0}}(y ⊕ p_{0}) & = (U_{x}^{J}y + ω_{p_{0}}(m)e) + (αν_{q_{0}}(n) + ν_{q_{0}}ν_{p_{0}}(m)) ⊕ U_{q_{0}}^{Q_{0}}(p_{0}), \\ (x ⊕ q_{0})^{2} & = x^{2} + ν_{q_{0}}(m) ⊕ q_{0}^{2} \end{aligned}

for x = αe ⊕ m, y = βe ⊕ n ∈ J, q_{0}, p_{0} ∈ Q_{0}, where the Peirce 0-component Q_{0} relative to e is a Jordan algebra with a linear specialization ν : Q_{0} → End(M) and a linear map ω : Q_{0} → WQ_{Φ}(M) satisfying (Axiom 1) ν_{U_{q_{0}} p_{0}} = ν_{q_{0}}ν_{p_{0}}ν_{q_{0}}, ν_{q_{0}^{2}} = ν_{q_{0}}ν_{q_{0}},

(Axiom 2) ω_{U_{q_{0}} p_{0}}(m) = ω_{p_{0}}(ν_{q_{0}}(m)), ω_{q_{0}^{2}} = 0

(hence ω_{{q_{0},p_{0},s_{0}}}(m) = ω_{p_{0}}(ν_{q_{0}}(m), ν_{s_{0}}(m)) = 0, ω_{q_{0}∘p_{0}} = 0),

(Axiom 3) ω_{q_{0}} = 0 and ν_{q_{0}} = 0 ⟹ q_{0} = 0.

The algebra Q is unital iff Q_{0} is unital, ν_{1_{Q_{0}}} = Id_{M}, and ω_{1_{Q_{0}}} (M) = 0. In this case 1_{Q} = e_{2} + e_{0} with e_{2} := e and e_{0} := 1_{Q_{0}}.

(2) There is a maximum algebra of quotients Q^{max} := J ⊕ Q_{0}^{max} for Q_{0}^{max} := End(M)^{(+)} ⊕ WQ_{Φ}(M) under

U_{T_{0}⊕τ}^{max} (S_{0} ⊕ σ) = T_{0}S_{0}T_{0} ⊕ σT_{0}, (T_{0} ⊕ τ)^{(2,max)} = T_{0}^{2} ⊕ 0

with ν^{max}, ω^{max} defined by

ν_{T_{0}⊕τ}^{max} := T_{0}, ω_{T_{0}⊕τ}^{max} := τ

and having unit e_{0} := Id_{M} ⊕ 0. Any quotient Q imbeds in this Q^{max} via φ = Id_{J} ⊕ ν ⊕ ω :

φ(x ⊕ q_{0}) = x ⊕ (ν_{q_{0}} ⊕ ω_{q_{0}}).

(3) In particular, if Φ = ℤ[ε] for a 2-dual number ε [2ε = ε^{2} = 0], and M = ℤm (εm = 0) then End_{Φ}(M) = ℤe_{0}, e_{0} = Id_{M}, RS(Φ) = ℤε = ℤ_{2}ε, WQ_{Φ}(M) = ℤεω_{0} = ℤ_{2}εω_{0} for ω_{0}(αm) = α^{2} the natural quadratic form on M, and J = Φe_{2} ⊕ M has maximum quotient Q^{max} = J ⊕ Q_{0}^{max} for Q_{0}^{max} = (ℤe_{0})^{(+)} ⊕ ℤεω_{0} = (ℤe_{0})^{(+)} ⊕ ℤ_{2}εω_{0}. In terms of direct sums of ℤ-modules J = ℤe_{2}⊕(ℤε)e_{2}⊕ℤm = ℤe_{2}⊕(ℤ_{2}ε)e_{2}⊕

ℤm, Q^{max} ≅ ( ℤ ℤ ℤ^{0}) ⊕ ℤ_{2}εe_{2} ⊕ ℤ_{2}εω_{0}e_{0} and Q_{0}^{max} = ℤ[εω_{0}]e_{0} = ℤe_{0} ⊕ (ℤεω_{0})e_{0}.

Proof: (1) (Necessity) Assume Q is a Martindale quotient of (J, F). It can be readily seen that Q results only from addition of a Peirce 0-component relative to e = e_{2}: if q = q_{0} + q_{1} + q_{2} ∈ Q (using subscripts to indicate the Peirce components) then q_{2} = U_{e}q ∈ U_{J}q ⊆ J, q_{1} = V_{e}(q) − 2q_{2} ∈ V_{J}q + J ⊆ J. Set ν_{q_{0}} := V_{q_{0}} |_{M} and ∩_{q_{0}}(m) = U_{m}q_{0} = ω_{q_{0}}(m)e_{2} and let E_{i} (i = 0, 1, 2) denote the Peirce projections with respect to e_{2}. Then Peirce orthogonality and triviality of M shows that the product in Q is given by the above Product Formula (use (0.2)(i) and notice that U_{M,M}Q_{0} = E_{2}(U_{M,M}Q_{0}) = E_{2}((M ∘ Q_{0}) ∘ M) ⊆ E_{2}(M ∘ M) = 0). To see that ω_{q_{0}} maps to RS_{Φ}(M), observe that the scalar λ := ω_{q_{0}}(m) satisfies λe_{2} = U_{m}q_{0} by definition, hence 2λe_{2} = 2U_{m}q_{0} = U_{m,m}q_{0} = 0; λ^{2}e_{2} = (U_{m}q_{0})^{2} = U_{m}U_{q_{0}}m^{2} = 0 [by (0.2)(iv) and M^{2} = 0], and λn = n ∘ U_{m}q_{0} = −U_{m}(n ∘ q_{0}) = 0, for all n ∈ M [by (0.2)(v) and M ∘ M = U_{M}M = 0]. To see that ω_{q_{0}} ∈ WQ_{Φ}(M), note that U_{M,M}q_{0} = 0 implies ω_{q_{0}}(M, M) = 0.

Then ν is the usual linear Peirce specialization of the Peirce 0-space on the Peirce 1-space, so Axiom 1 holds. For Axiom 2, ω_{q_{0}^{2}}(m)e_{2} = U_{m}(q_{0}^{2}) = 0 by (0.2)(iii), and then ω_{U_{q_{0}} p_{0}}(m)e_{2} = U_{m∘q_{0}}p_{0} = ω_{p_{0}}(V_{q_{0}}m)e_{2}. Axiom 3 is the necessary and sufficient condition for Q to be a Martindale quotient: always β_{J}(q) ⊆ J, and β_{J}(q) = 0 ⟺ q = q_{0}, V_{q_{0}}|_{M} = 0, ∩_{q_{0}}|_{M} = 0 ⟺ q = q_{0}, ν_{q_{0}} = 0, ω_{q_{0}} = 0. Thus the Axioms are necessary.

If Q has a unit 1 (which means U_{1} = Id_{Q} and U_{q}1 = q^{2} for all q ∈ Q), it is readily seen that 1 = e_{2} + e_{0} for some e_{0} ∈ Q_{0}. A direct computation shows that U_{1} = Id_{Q} is equivalent to U_{e_{0}} = E_{0}, U_{e_{2},e_{0}} = E_{1}, while q^{2} = U_{q}1 is equivalent to q_{0}^{2} = U_{q_{0}}e_{0}, ω_{e_{0}} = 0, and V_{q_{0},e_{0}}|_{M} = V_{q_{0}}|_{M}. From this, the above criterion of unitality of Q readily follows, taking into account that always U_{e_{0},e_{2}} = V_{e_{0}}E_{1}.

(Sufficiency) It is more tedious to prove that Axioms 1,2 and the Product Formula are sufficient to produce a quadratic Jordan algebra. For convenience we pass to the free unital hull Q̂_{0} = Φe_{0} ⊕ Q_{0} (where Axiom 3 might not hold any longer) with linear specialization ν̂ of Q̂_{0} via ν̂_{αe_{0}+q_{0}} := αId_{M} + ν_{q_{0}} satisfying obviously Axiom 1, and quadratic ω̂_{αe_{0}⊕q_{0}} := ω_{q_{0}} satisfying Axiom 2 since ω̂_{U_{αe_{0}+q_{0}} βe_{0}+p_{0}} = α^{2}ω_{p_{0}} + ω_{U_{q_{0}} p_{0}} = ω_{p_{0}} · (αId_{M} + ν_{q_{0}}) = ω̂_{βe_{0}+p_{0}} ν̂_{αe_{0}+q_{0}} (using that ω_{q_{0}} maps to RS_{Φ}(M)) and ω_{(αe_{0}+q_{0})^{2}} = ω_{2αq_{0}+q_{0}^{2}} = 0. We will assume from the start that Q_{0} is unital (with unit e_{0} satisfying ν_{e_{0}} = Id_{M}, ω_{e_{0}}(M) = 0), and verify the quadratic axioms (QJA1-3).

By definition of WQ_{Φ}(M), the Product Formula, and Axioms 1,2 we have

\begin{matrix}(⋆) & V_{M}U_{M}E_{0} = 2U_{M}E_{0} = U_{M,M}E_{0} = U_{M}V_{Q_{0},Q_{0}}E_{0} = ω_{Q_{0}}(M)E_{1} = 0, \\ (⋆⋆) & V_{q_{0},m_{1}}E_{2} = V_{q_{0}∘m_{1}}E_{2} = V_{q_{0}}V_{m_{1}}E_{2}, V_{m_{1},q_{0}}E_{0} = V_{q_{0}∘m_{1}}E_{0}. \end{matrix}

Unitality (QJA1) U_{1} = Id_{Q} for 1 = e_{2} + e_{0} follows from the Product Formula. To establish (QJA2-3) we must look carefully at the operators involving general elements q := αe_{2} ⊕ m_{1} ⊕ q_{0}, p := βe_{2} ⊕ n_{1} ⊕ p_{0}: by the Product Formula U_{q} = α^{2}U_{e_{2}}+U_{m_{1}}+U_{q_{0}}+αU_{e_{2},m_{1}}+αU_{e_{2},q_{0}}+U_{m_{1},q_{0}} = ∑_{i,j} X_{ij} =: X for Peirce components X_{ij} := E_{i}U_{q}E_{j}, and similarly U_{p} = ∑_{i,j} Y_{ij} =: Y, V_{q,p} = ∑_{i,j} S_{ij} =: S, V_{p,q} = ∑_{i,j} T_{ij} =: T where the Peirce components are given by

\begin{aligned}X_{21} & = X_{02} = X_{01} = 0, X_{22} = α^{2}E_{2}, X_{00} = U_{q_{0}} = U_{q_{0}}E_{0}, X_{11} = αV_{q_{0}}E_{1}, \\ X_{20} & = U_{m_{1}}E_{0}, X_{12} = αV_{m_{1}}E_{2}, X_{10} = V_{q_{0}}V_{m_{1}}E_{0}, \\ Y_{21} & = Y_{02} = Y_{01} = 0, Y_{22} = β^{2}E_{2}, Y_{00} = U_{p_{0}} = U_{p_{0}}E_{0}, Y_{11} = βV_{p_{0}}E_{1}, \\ Y_{20} & = U_{n_{1}}E_{0}, Y_{12} = βV_{n_{1}}E_{2}, Y_{10} = V_{p_{0}}V_{n_{1}}E_{0}, \\ S_{01} & = S_{21} = S_{02} = S_{20} = 0, S_{22} = 2αβE_{2}, S_{11} = (αβ + V_{q_{0}}V_{p_{0}})E_{1}, \\ S_{00} & = V_{q_{0},p_{0}}E_{0}, S_{10} = (αV_{n_{1}} + V_{p_{0}∘m_{1}})E_{0}, S_{12} = (βV_{m_{1}} + V_{q_{0}}V_{n_{1}})E_{2}, \\ T_{01} & = T_{21} = T_{02} = T_{20} = 0, T_{22} = 2βαE_{2}, T_{11} = (βα + V_{p_{0}}V_{q_{0}})E_{1}, \\ T_{00} & = V_{p_{0},q_{0}}E_{0}, T_{10} = (βV_{m_{1}} + V_{q_{0}∘n_{1}})E_{0}, T_{12} = (αV_{n_{1}} + V_{p_{0}}V_{m_{1}})E_{2} \end{aligned}

[using (⋆) (⋆⋆) for S_{ij}, T_{ij}]. To establish (QJA2), SX − XT = 0, we check that E_{i}(SX − XT)E_{j} = 0 directly, using (⋆) (⋆⋆), for all (i, j) except (i, j) = (0, 0), where we use (QJA2) for Q_{0}, and (1, 0), where we use V_{q_{0}}V_{m_{1}}V_{p_{0},q_{0}}s_{0} = V_{q_{0}}V_{{p_{0},q_{0},s_{0}}}m_{1} = V_{q_{0}}(V_{p_{0}}V_{q_{0}}V_{s_{0}} +V_{s_{0}}V_{q_{0}}V_{p_{0}})m_{1} [linearizing Axiom 1] = (V_{q_{0}}V_{p_{0}}V_{q_{0}}V_{m_{1}} +V_{p_{0}∘m_{1}}U_{q_{0}})s_{0} and V_{q_{0}}V_{q_{0}∘n_{1}}s_{0} = V_{q_{0}}V_{s_{0}}V_{q_{0}}n_{1} = V_{U_{q_{0}} s_{0}}n_{1} [by Axiom 1] = V_{n_{1}}U_{q_{0}}s_{0}. This completes the proof of (QJA2).

Finally, for the Fundamental Formula (QJA3), U_{U_{q}p} = U_{q}U_{p}U_{q} = XY X, we have by the Product Formula

\begin{aligned}s & := U_{q}p = γe_{2} + r_{1} + s_{0} for γ := α^{2}β + ω_{p_{0}}(m_{1}), γ^{2} = α^{4}β^{2}, \\ s_{0} & := U_{q_{0}}p_{0}, r_{1} := αβm_{1} + αq_{0} ∘ n_{1} + V_{q_{0}}V_{p_{0}}m_{1} \end{aligned}

so U_{s} = ∑_{i,j} Z_{ij} =: Z has Peirce components

Z_{21} = Z_{02} = Z_{01} = 0, Z_{22} = α^{4}β^{2}E_{2},
Z_{00} = U_{s_{0}} = U_{q_{0}}U_{p_{0}}U_{q_{0}}, Z_{11} = γV_{s_{0}}E_{1} = α^{2}βV_{q_{0}}V_{p_{0}}V_{q_{0}}E_{1}, [by (QJA3), Ax 1,⋆]
Z_{12} = γV_{r_{1}}E_{2} = α^{3}β^{2}V_{m_{1}}E_{2}+α^{3}βV_{q_{0}}V_{n_{1}}E_{2}+α^{2}βV_{q_{0}}V_{p_{0}}V_{m_{1}}E_{2},  [by ⋆, ⋆⋆]
Z_{20} = U_{r_{1}}E_{0} = α^{2}β^{2}U_{m_{1}}E_{0}+α^{2}U_{n_{1}}U_{q_{0}}E_{0}+U_{m_{1}}U_{p_{0}}U_{q_{0}}E_{0},  [by ⋆, Ax 2]
Z_{10} = V_{s_{0}}V_{r_{1}}E_{0} = V_{q_{0}}V_{p_{0}}V_{q_{0}}(αβV_{m_{1}} + αV_{q_{0}∘n_{1}} + V_{V_{q_{0}} V_{p_{0}} m_{1}})E_{0}.  [by Ax 1]

By direct calculation the Z_{ij} are the same as the components E_{i}XY XE_{j} of U_{q}U_{p}U_{q}: E_{2}XY XE_{1} = E_{0}XY XE_{2} = E_{0}XY XE_{1} = 0, E_{2}XY XE_{2} = Z_{22}, E_{0}XY XE_{0} = Z_{00}, E_{1}XY XE_{1} = Z_{11}, E_{1}XY XE_{2} = Z_{12}, E_{2}XY XE_{0} = Z_{20}, E_{1}XY XE_{0} = Z_{00} [using Ax 1, ⋆]. Thus (QJA3) holds.

Axioms 1-2 hold strictly (on all scalar extensions), so the identities (QJA1-3) hold strictly. Once we have proved that Q is a Jordan algebra we return to the general case where Q_{0} might not be unital, but Axiom 3 holds. From this axiom and the Product Formula it is readily checked that (Q, τ) is a Martindale quotient of (J, F), which finishes the verification (1).

(2) It is easy to verify that Q_{0}^{max} is a Jordan algebra which satisfies the 3 axioms: (QJA1-3) are straightforward; since any product {x, y, z} in Q_{0}^{max} equals the corresponding product of the first components of x, y, z in End(M)^{(+)}, the linearizations of (QJA1-3) also hold. For q_{0} = T_{0} ⊕ τ, p_{0} = S_{0} ⊕ σ we have Axiom 1 since ν_{U_{q_{0}} p_{0}} = ν_{T_{0}S_{0}T_{0}⊕σ∘T_{0}} = T_{0}S_{0}T_{0} = ν_{q_{0}}ν_{p_{0}}ν_{q_{0}}, ν_{q_{0}^{2}} = ν_{T_{0}^{2}⊕0} = T_{0}T_{0} = ν_{q_{0}}ν_{q_{0}}, while Axiom 2 holds because ω_{U_{q_{0}} p_{0}} = ω_{T_{0}S_{0}T_{0}⊕σT_{0}} = σT_{0} = ω_{p_{0}}ν_{q_{0}} and ω_{q_{0}^{2}} = ω_{T_{0}^{2}⊕0} = 0, and Axiom 3 follows from ω_{q_{0}} = 0 ⇒ τ = 0 and ν_{q_{0}} = 0 ⇒ T_{0} = 0. Q_{0}^{max} has unit e_{0} = Id_{M} ⊕ 0 and ν_{e_{0}} = Id_{M} and ω_{e_{0}} = 0 as in the unitality criterion, hence Q^{max} is unital and a Martindale quotient of (J, F) with multiplication given by the Product Formula by (1).

For the universal imbedding property of Q in Q^{max}, the linear map φ is injective since Ker(φ) = {q_{0} ∈ Q_{0} | ν_{q_{0}} = 0, ω_{q_{0}} = 0} vanishes by Axiom 3, and it is a homomorphism of Jordan algebras since both have Product Formulas, maps ν, ω, and algebras Q_{0}, Q_{0}^{max} which correspond under φ:

ν_{q_{0}} = ν_{ν_{q_{0}} ⊕ω_{q_{0}}}^{max} = ν_{φ(q_{0})}^{max},  ω_{q_{0}} = ω_{ν_{q_{0}} ⊕ω_{q_{0}}}^{max} = ω_{φ(q_{0})}^{max},
φ(U_{q_{0}}p_{0}) = ν_{U_{q_{0}} p_{0}} ⊕ ω_{U_{q_{0}} p_{0}} = ν_{q_{0}}ν_{p_{0}}ν_{q_{0}} ⊕ ω_{p_{0}}ν_{q_{0}}  [by Ax 1,2]
= U_{ν_{q_{0}} ⊕ω_{q_{0}}}^{max} (ν_{p_{0}} ⊕ ω_{p_{0}}) = U_{φ(q_{0})}^{max}(φ(p_{0})),
φ(q_{0}^{2}) = ν_{q_{0}^{2}} ⊕ ω_{q_{0}^{2}} = ν_{q_{0}}ν_{q_{0}} ⊕ 0[by Ax 1,2] = (ν_{q_{0}} ⊕ ω_{q_{0}})^{(2,max)} = φ(q_{0})^{(2,max)}.

This establishes that Q^{max} is a maximum quotient.

(3) This follows immediately from (2) since λ = α + βε ∈ ℤ[ε] has λM = 0 and λ^{2} = 2λ = 0 if and only if α = 0, and ω(m) = βε ⇒ ω(αm) = α^{2}βε = βεω_{0}(αm) ⇒ ω = βεω_{0}.

When imbedded in Q^{max}, the quotient Q need not split into a direct sum of three components, but it does contain an ideal W_{0} := {q_{0} ∈ Q_{0} | ν_{q_{0}} = 0} = Ker(ν) of Q_{0} (by Axiom 1) and an ample outer ideal E_{0} := {q_{0} ∈ Q_{0} | ω_{q_{0}} = 0} of Q_{0} with E_{0} ⊕ W_{0} ⊆ Q_{0} which is a direct sum of subspaces (though not of algebras) by Axiom 3. [For outerness, E_{0} is invariant under all U_{p_{0}} and all V_{p_{0}} since ω_{U_{p_{0}} q_{0}} = ω_{q_{0}}ν_{p_{0}} = 0 and ω_{p_{0}∘q_{0}} = 0 by Axiom 2, and for ampleness, we use again Axiom 2 together with ω_{Q_{0}}(M) ⊆ RSΦ(M)].

4. Building Martindale Quotients out of Extensions

It is important that once we can boost an element q into J, we can boost it into any ideal K of J we wish.

4.1 Lemma. Let J be a subalgebra of a Jordan algebra Q, and q ∈ Q an element boosted into J by an ideal I of J. Then for any other ideal K of J, the cube of the ideal I′ := (I ∩ K)^{3} boosts q into K: if β_{I}(q) ⊆ J then

(i) β_{I′}^{3}(q) ⊆ U_{I′}q + U_{q}I′^{3} + q ∘ I′ ⊆ K,
(ii) V_{I′^{3},q} + V_{q,I′^{3}} ⊆ V_{K,K}.

Proof: To establish (i), U_{I′}q + q ∘ I′ ⊆ K since for k = U_{a}b ∈ I′ (a, b ∈ I ∩ K), k′ ∈ I′ we have U_{k}q = U_{a}U_{b}U_{a}q [by (0.2)(iv)] ∈ U_{K}U_{K}(U_{I}q) ⊆ U_{K}U_{K}J ⊆ K and U_{k,k′}q, k ∘ q ∈ V_{I′,q}Ĵ ⊆ K since

(iii) V_{I′,q} + V_{q,I′} ⊆ V_{I∩K,J} + V_{J,I∩K} ⊆ V_{K,J} + V_{J,K}

by (1.1)(i) for I ∩ K in place of I, and S = J. Finally, U_{q}I′^{3} ⊆ K since for a, b ∈ I′ ⊆ I^{3} ∩ K (0.2)(iii) implies U_{q}U_{a}b = (U_{q∘a} − U_{a}U_{q} − V_{q,a}V_{a,q} + V_{U_{q}a^{2}})b ∈ U_{q∘I}K + U_{K}(U_{q}I) + V_{q,I′}V_{I′,q}K + V_{U_{q}I}K ⊆ U_{J}K + U_{K}J + (V_{K,J} + V_{J,K})^{2}K + V_{J}K [by (iii)] ⊆ K.

(ii) follows from (i) since U_{I′}q+q∘I′ ⊆ K implies V_{I′^{3},q} +V_{q,I′^{3}} ⊆ V_{I′,K} +V_{K,I′} ⊆ V_{K,K} by (1.1)(i) [I′ in place of I, S = K].

Note that if I, K ∈ F for a filter F, then we can choose L ∈ F such that L ⊆ I^{′3} := ((I ∩ K)^{3})^{3}, so that L boosts q into K.

4.2 Theorem. Let f : J ⟶ Q be a homomorphism of Jordan algebras and F be a filter of ideals of J.

(i) The boostable elements

Q(f, F) := {q ∈ Q | ∃I ∈ F with β_{f(I)}(q) ⊆ f(J)}

of Q form a Jordan subalgebra of Q containing f(J) ( and the unit of Q if Q is unital).

(ii) The filter annihilator

Ann(f, F) : = {q ∈ Q(f, F) | ∃I ∈ F with β_{f(I)}(q) = 0}
= {q ∈ Q(f, F) | ∃I ∈ F with q ∈ Zann_{Q}(f(I))}

(the elements boosted to 0 by some ideal in the filter) is an ideal of Q(f, F).

(iii) If all the ideals f(I), for I ∈ F, are sturdy in f(J) (for example, if F is a denominator filter of J and f is injective) then

(a) Ann(f, F) ∩ f(J) = 0,

(b) if q ∈ Q(f, F) then β_{f(I)}(q) ⊆ Ann(f, F), for some I ∈ F, implies

q ∈ Ann(f, F).

If, in addition, f is injective, then it induces the algebra monomorphism

(c) f̃: J ⟶ Q̃(f, F) := Q(f, F)/ Ann(f, F)

given by f̃(x) = f(x) + Ann(f, F). Moreover, (J, F) is a denominatored algebra and (Q̃(f, F), f̃) is a Martindale quotient of (J, F).

Proof: Notice that F′ = {f(I) | I ∈ F} is a filter of ideals of the subalgebra f(J) of Q, so Q(f, F) = Q(τ, F′), Ann(f, F) = Ann(τ, F′), where τ : f(J) ⟶ Q is the inclusion map. We will henceforth assume that J is a subalgebra of Q and f is the inclusion (we are not assuming our new J is the same as the old since f need not be a monomorphism).

(i) Imbedding Q in a unital hull Q̂, it suffices to prove the unital version (this guarantees Q(f, F) is closed under squares q^{2} = U_{q}1 as soon as it is closed under U-products U_{q}q′ - note that always 1 ∈ Q̂(f, F) since β_{I}(1) ⊆ I ⊆ J for all I ∈ F).

Throughout we fix q_{1}, q_{2} ∈ Q(f, F) boosted by the ideals I_{1}, I_{2} ∈ F. Since F is a filter, we can find I ∈ F with I ⊆ I_{1} ∩ I_{2} a common booster for q_{1} and q_{2}. We also fix L, M ∈ F with L ⊆ (I^{3})^{3} and M ⊆ (L^{3})^{3}.

Let q = q_{1} or q_{2}. From (1.1)(i) for S = J, we obtain
{I, I, q} ⊆ J.  (1)

If, in addition, we use (4.1) for K = I (so I′ = I^{3}) or K = L (so I′ = (I ∩ L)^{3} = L^{3}), we obtain respectively,

U_{I^{3}}q + q ∘ I^{3} + U_{q}L ⊆ I, V_{L,q} + V_{q,L} ⊆ V_{I,I}  (2)

or

U_{L^{3}}q + q ∘ L^{3} + U_{q}M ⊆ L, V_{M,q} + V_{q,M} ⊆ V_{L,L}.  (3)

First, αq ∈ Q(f, F) for any α ∈ Φ since β_{I}(αq) ⊆ αU_{I}q + α^{2} ∩_{I} q + αV_{I}q ⊆ J. Next, q_{1}+q_{2} ∈ Q(f, F) since β_{L}(q_{1}+q_{2}) ⊆ J: (U_{L}+V_{L})(q_{1}+q_{2}) ⊆ (U_{I}+V_{I})q_{1}+(U_{I}+V_{I})q_{2} ⊆ J and ∩_{L}(q_{1} + q_{2}) ⊆ ∩_{L}q_{1} + ∩_{L}q_{2} + {q_{1}, L, q_{2}} ⊆ J + J + V_{q_{1},L}q_{2} ⊆ J + V_{I,I}q_{2} (by (2)) ⊆ J by (1). Finally U_{q_{1}}q_{2} ∈ Q(f, F) since β_{M}(U_{q_{1}}q_{2}) ⊆ J: V_{M}(U_{q_{1}}q_{2}) ⊆ {q_{1} ∘ M, q_{2}, q_{1}} + U_{q_{1}}(M ∘ q_{2}) (by (0.2)(v)) ⊆ V_{L,q_{2}}q_{1} + U_{q_{1}}L (by (3)) ⊆ V_{I,I}q_{1} + I (by (2)) ⊆ J by (1); U_{U_{q_{1}} q_{2}}M = U_{q_{1}}U_{q_{2}}U_{q_{1}}M (by (0.2)(iv)) ⊆ U_{q_{1}}U_{q_{2}}L (by (3)) ⊆ U_{q_{1}}I (by (2)) ⊆ J; and U_{L}(U_{q_{1}}q_{2}) ⊆ U_{L∘q_{1}}q_{2} + U_{q_{1}}(U_{L}q_{2}) + V_{q_{1},L}V_{L,q_{1}}q_{2} + V_{U_{q_{1}} L^{2}}q_{2} (by (0.2)(iii)) ⊆ U_{I}q_{2} + U_{q_{1}}I + V_{I,I}V_{I,I}q_{2} + V_{I}q_{2} (by (2)) ⊆ J by (1).

(ii) Note that the two conditions for q ∈ A := Ann(f, F) are equivalent using (1.1)(ii): if β_{I}(q) = 0 then q ∈ Zann_{Q}(I^{3}), and q ∈ Zann_{Q}(I′) for any I′ ∈ F with I′ ⊆ I^{3}. As in (i), we may assume that Q is unital. To see that A is an ideal, we again consider q_{1}, q_{2} ∈ A, p ∈ Q(f, F) which we may assume have a common booster I ∈ F such that β_{I}(q_{i}) = 0 (i = 1, 2), β_{I}(p) ⊆ J, where again by (1.1)(i) with S = 0 or J,

{I, I, q_{i}} = 0,  for i = 1, 2,  or  {I, I, p} ⊆ J,  (4)

and the relations (1–3) still hold for q = q_{1}, q_{2}, p for L, M ∈ F as above. Also, by (1.1)(i) with S = 0,

V_{I^{3},q_{i}} = V_{q_{i},I^{3}} = 0  for i = 1, 2.  (5)

Clearly A is closed under scaling, and it is closed under sums since as in (i) above β_{I^{3}}(q_{1} + q_{2}) = 0: (U_{I} + V_{I})(q_{1} + q_{2}) ⊆ (U_{I} + V_{I})q_{1} + (U_{I} + V_{I})q_{2} = 0, and ∩_{I^{3}}(q_{1} + q_{2}) ⊆ ∩_{I^{3}}q_{1} + ∩_{I^{3}}q_{2} + V_{q_{1},I^{3}}q_{2} = 0 by (5).

By unitality, A will be an ideal as soon as all U_{p}q_{1} and U_{q_{1}}p lie in A. Put q = q_{1}. For U_{p}q, we have β_{M}(U_{p}q) = 0: V_{M}(U_{p}q) = {p ∘ M, q, p} + U_{p}(M ∘ q) (by (0.2)(v)) ⊆ V_{L,q}p + U_{p}(I ∘ q) (by (3)) = 0 by (5); ∩_{L}(U_{p}q) = U_{p}U_{q}U_{p}L (by (0.2)(iv)) ⊆ U_{p}U_{q}I (by (2)) = 0; and U_{L}(U_{p}q) ⊆ U_{L∘p}q + U_{p}U_{L}q + V_{p,L}V_{L,p}q + V_{U_{p}L^{2}}q (by (0.2)(iii)) ⊆ U_{I}q + U_{p}(U_{I}q) + V_{I,I}V_{I,I}q + V_{I}q (by (2)) = 0 by (4). For U_{q}p we have β_{I^{3}}(U_{q}p) = 0: V_{I^{3}}(U_{q}p) ⊆ {I^{3} ∘ q, p, q} + U_{q}(I^{3} ∘ p) (by (0.2)(v)) ⊆ {I ∘ q, p, q} + U_{q}I (by (2)) = 0; ∩_{I}(U_{q}p) = U_{q}U_{p}U_{q}I (by (0.2)(iv)) = 0; and U_{I^{3}}(U_{q}p) = U_{I^{3}∘q}p + U_{q}U_{I^{3}}p + V_{q,I^{3}}V_{I^{3},q}p + V_{U_{q}(I^{3})^{2}}p (by (0.2)(iii)) ⊆ U_{I∘q}p + U_{q}I + 0 + V_{U_{q}I}p (by (2), (5)) = 0.

(iii) Now assume that all I ∈ F are sturdy in J.

(a) If x ∈ J ∩ Ann(f, F), then there exists L ∈ F such that x ∈ Zann_{Q}(L) ∩ J ⊆ Zann_{J}(L) = 0 by sturdiness.

(b) Let q ∈ Q(f, F), so that there exists L ∈ F such that β_{L}(q) ⊆ J and suppose that there is I ∈ F, such that β_{I}(q) ⊆ Ann(f, F). Let K ∈ F satisfy K ⊆ I ∩ L. Hence, (a) yields β_{K}(q) ⊆ Ann(f, F) ∩ J = 0, which implies q ∈ Ann(f, F).

The rest of (iii) is straightforward.

5. Existence of Maximal Martindale Quotients

5.1 We let Mq(J, F) denote the class of Martindale quotients of a denominatored algebra (J, F). If (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) ∈ Mq(J, F) we will say that (Q_{1}, τ_{1}) is less than or equal to (Q_{2}, τ_{2}), and write (Q_{1}, τ_{1}) ≤ (Q_{2}, τ_{2}), if there exists an algebra homomorphism f : Q_{1} ⟶ Q_{2} such that τ_{2} = fτ_{1}. By (2.4), any such covering f is actually a monomorphism. We say that (Q_{1}, τ_{1}) is isomorphic to (Q_{2}, τ_{2}), and write (Q_{1}, τ_{1}) ≅ (Q_{2}, τ_{2}) if there exists an algebra isomorphism f : Q_{1} ⟶ Q_{2} such that fτ_{1} = τ_{2}. [Q, τ] will denote the class of all Martindale quotients of (J, F) isomorphic to (Q, τ). A Martindale quotient (Q, τ) ∈ Mq(J, F) will be said to be maximal if any other (Q′, τ′) ∈ Mq(J, F) bigger than or equal to (Q, τ) is necessarily isomorphic to it: (Q, τ) ≤ (Q′, τ′) ⟹ (Q, τ) ≅ (Q′, τ′).

A priori, the collection of isomorphism classes of quotients of (J, F) form a class; we wish to show they can be fully represented by a set, indeed a partially ordered set.

5.2 Proposition. Given a denominatored algebra (J, F), there is a bound on the cardinalities of all Martindale quotients of (J, F). Moreover, every Martindale quotient is isomorphic to an algebra based on a subset of the fixed set

X(J, F) = ⊎_{I∈F}X_{I}, X_{I} := Quad_{Φ}(I, J) × Hom_{Φ}(I, J) × Hom_{Φ}(I, J),

where ⊎ denotes the disjoint union, Quad_{Φ}(I, J) denotes the set of quadratic maps from I to J, and Hom_{Φ}(I, J) denotes the set of Φ-linear maps from I to J.

Proof: Given a Martindale quotient (Q, τ) of (J, F) we can, as usual, assume that J ⊆ Q and τ is inclusion. We define a set-theoretic map φ : Q ⟶ X := X(J, F), by choosing for each q ∈ Q an ideal I_{q} ∈ F such that β_{I_{q}}(q) ⊆ J, and then define φ(q) := (∩_{q}, U_{q}, V_{q}) ∈ X_{I_{q}} ⊆ X(J, F). We claim that φ is injective. Indeed, if φ(q) = φ(q′), then I_{q} = I_{q′} =: I ∈ F has

∩_{q} = ∩_{q′}, V_{q} = V_{q′} on I ⟹ U_{I}(q − q′) = V_{I}(q − q′) = 0,

hence V_{q−q′,I^{3}} = 0 by (1.1)(i) for S = 0. Thus, for any x ∈ I^{3}, ∩_{x}(q − q′) = U_{q−q′}x = U_{q}x+U_{q′}x−{q, x, q′} = 2U_{q}x−{q, x, q′} [since U_{q} = U_{q′} on I] = {q, x, q}−{q, x, q′} = {q, x, q − q′} ∈ V_{q−q′,I^{3}}Q = 0, so ∩_{I^{3}}(q − q′) = 0. Therefore β_{K}(q − q′) = 0 for any K ∈ F such that K ⊆ I^{3}, hence by (2.6)(i) q − q′ = 0 and q = q′. By set-theoretic transfer, the bijection Q ⟶ φ(Q) ⊆ X becomes an isomorphism of Q with an algebra Q′ based on a subset of X.

5.3 Let Mq(J, F)_{X} denote the set of all Martindale quotients (Q, τ) based on subsets of X = X(J, F), and Iso_{X}(J, F) be the quotient of Mq(J, F)_{X} by the restriction of the isomorphism relation, i.e., the set of isomorphism classes [Q, τ]_{X} of the quotients (Q, τ) ∈ Mq(J, F)_{X}. (The X-class [Q, τ]_{X} is not all of [Q, τ], it contains only those isomorphic algebras based on subsets of X.) By (5.2) every Martindale quotient (Q′, τ′) is isomorphic to some (Q, τ) ∈ Mq(J, F)_{X}, and we can define the X-representative [Q′, τ′]_{X} := [Q, τ]_{X} ∈ Iso_{X}(J, F) which is independent of our particular choice of the isomorphic Q. By abuse of language we say the set Iso_{X}(J, F) represents all isomorphism classes of Martindale quotients. Notice that, for (Q, τ), (Q′, τ′) ∈ Mq(J, F),

(Q, τ) ≅ (Q′, τ′) ⟺ [Q, τ] = [Q′, τ′] ⟺ [Q, τ]_{X} = [Q′, τ′]_{X}.

5.4 Proposition. For any denominatored algebra (J, F), we can define a binary relation ≤ on Iso_{X}(J, F) given by [Q_{1}, τ_{1}]_{X} ≤ [Q_{2}, τ_{2}]_{X} if (Q_{1}, τ_{1}) ≤ (Q_{2}, τ_{2}) so that (Iso_{X}(J, F), ≤) is a nonempty inductive partially ordered set and hence contains maximal elements. Therefore Mq(J, F) contains maximal elements, and any element (Q, τ) of Mq(J, F) is less than or equal to a maximal element of Mq(J, F).

Proof: Iso_{X}(J, F) is nonempty because (J, Id_{J}) ∈ Mq(J, F) by (2.5). The relation ≤ is well-defined on isomorphism classes since it doesn’t depend on class representatives: f : Q_{1} ⟶ Q_{2} covering τ_{1}, τ_{2} induces f′ : Q′_{1} ⟶ Q′_{2} covering τ′_{1}, τ′_{2} for any quotients (Q′_{i}, τ′_{i}) ≅ (Q_{i}, τ_{i}). Moreover we have

(1)  (Q_{1}, τ_{1}) ≤ (Q_{2}, τ_{2}) in Mq(J, F) ⟺ [Q_{1}, τ_{1}]_{X} ≤ [Q_{2}, τ_{2}]_{X} in Iso_{X}(J, F).

To see ≤ is a partial ordering, it is clearly reflexive and transitive, and it is antisymmetric on classes (not on individual algebras) since if [Q, τ]_{X} ≤ [Q′, τ′]_{X} ≤ [Q, τ]_{X} then there exist algebra homomorphisms f′ : Q ⟶ Q′ and f : Q′ ⟶ Q such that τ′ = f′τ and τ = fτ′. Hence, as in the proof of (2.9), f′fτ′ = τ′ = Id_{Q′}τ′ and ff′τ = τ = Id_{Q}τ imply f′f = Id_{Q′} and ff′ = Id_{Q} by uniqueness (2.7). This shows that (Q, τ) ≅ (Q′, τ′), i.e., [Q, τ]_{X} = [Q′, τ′]_{X} by (5.3).

To check inductiveness, let {[Q_{ι}, τ_{ι}]_{X} | ι ∈ S} be a chain in (Iso_{X}(J, F)_{X}, ≤): for any ι, κ ∈ S, either [Q_{ι}, τ_{ι}]_{X} ≤ [Q_{κ}, τ_{κ}]_{X} or [Q_{κ}, τ_{κ}]_{X} ≤ [Q_{ι}, τ_{ι}]_{X}. When [Q_{ι}, τ_{ι}]_{X} ≤ [Q_{κ}, τ_{κ}]_{X}, let f_{ικ} : Q_{ι} ⟶ Q_{κ} denote the unique (2.7) algebra monomorphism satisfying τ_{κ} = f_{ικ}τ_{ι}, so by uniqueness f_{ιλ} = f_{κλ} ∘ f_{ικ}. The direct limit of (Q_{ι}, f_{ικ}) will play the role of the “union” of the sets Q_{ι}’s to get a suitable upper bound. Indeed, Q = \underrightarrow{lim} Q_{ι} is a Jordan algebra, we have monomorphisms f_{ι} : Q_{ι} ⟶ Q synthesizing the f_{κι} : Q_{ι} ⟶ Q_{κ}, and they induce an algebra monomorphism τ : J ⟶ Q given by τ = f_{ι}τ_{ι}, for all ι ∈ S (recall that Q can be built as Q = ⊎_{ι∈S}Q_{ι}/R, where R denotes the binary relation in the disjoint union ⊎_{ι∈S}Q_{ι} given by xRf_{ικ}(x) when x ∈ Q_{ι} and (Q_{ι}, τ_{ι}) ≤ (Q_{κ}, τ_{κ}), which can be readily seen to be an equivalence relation). It is easy to see that (Q, τ) is a Martindale quotient of (J, F), and the very definition of τ shows that [Q, τ]_{X} is an upper bound for the chain {[Q_{ι}, τ_{ι}]_{X}}_{ι∈S} in (Iso_{X}(J, F), ≤). This guarantees the existence of maximal elements in (Iso_{X}(J, F), ≤) by Zorn’s Lemma, and, indeed, the fact that any element of Iso_{X}(J, F) is less than or equal to a maximal element. The assertions on maximal elements of Mq(J, F) follow from (1) and its elementary consequence for any (Q, τ) ∈ Mq(J, F):

(2) (Q, τ) is a maximal quotient ⟺ [Q, τ]_{X} is maximal in (Iso_{X}(J, F), ≤).

5.5 Remark: Notice that for a Martindale quotient (Q, τ) of (J, F), being maximum in the sense of (2.8) is just [Q, τ]_{X} being the maximum of (Iso_{X}(J, F), ≤). Note that the above direct limit Q need not be based on X even though the Q_{ι} are, but, by (5.3), [Q, τ]_{X} = [Q′, τ′]_{X} for an isomorphic quotient Q′ ∈ Mq(J, F)_{X}.

The construction of the previous section can be used to characterize the existence of maximum Martindale quotients.

5.6 Theorem. Let (J, F) be a denominatored algebra. The following assertions are equivalent:

(i) (Iso_{X}(J, F), ≤) has a maximum.

(ii) There exist a Jordan algebra Q and an algebra monomorphism f : J ⟶ Q such that for any (Q_{1}, τ_{1}) ∈ Mq(J, F), there exists an algebra homomorphism f_{1} : Q_{1} ⟶ Q, such that f_{1}τ_{1} = f.

(iii) For any (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) ∈ Mq(J, F), there exist a Jordan algebra Q, an algebra monomorphism f : J ⟶ Q, and algebra homomorphisms f_{i} : Q_{i} ⟶ Q, such that f_{i}τ_{i} = f for i = 1, 2.

(iv) (Iso_{X}(J, F), ≤) is directed: for any (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) ∈ Mq(J, F), there exists (Q, τ) ∈ Mq(J, F), such that (Q_{i}, τ_{i}) ≤ (Q, τ), for i = 1, 2, equivalently, there exists [Q, τ]_{X} ∈ Iso_{X}(J, F) such that [Q_{i}, τ_{i}]_{X} ≤ [Q, τ]_{X}.

Proof: (i)⟹(ii): If (Iso_{X}(J, F), ≤) has a maximum [Q, τ]_{X}, one just needs to take f = τ and the existence of f_{1} in (ii) will follow from equivalence (5.4)(1).

(ii)⟹(iii) is obvious.

(iii)⟹(iv): We can consider Q̃(f, F) of (4.2)(iii), so that f induces τ := f̃ : J ⟶ Q̃(f, F) and (Q̃(f, F), τ) ∈ Mq(J, F). Moreover, it can be readily seen that, for i = 1, 2, f_{i}(Q_{i}) ⊆ Q(f, F), so that f_{i} induces an algebra homomorphism g_{i} : Q_{i} ⟶ Q̃(f, F) such that g_{i}τ_{i} = τ, which shows (Q_{i}, τ_{i}) ≤ (Q̃(f, F), τ).

(iv)⟹(i): Being directed implies that there is at most one maximal element. This, together with (5.4), yields the existence of a maximum.

It is known [14] that each nondegenerate Jordan algebra has a maximum Martindale quotient algebra, but it is not known if this holds in general. However, it always has a minimum quotient (itself), and any two quotients have an infimum.

5.7 To define the infimum of two Martindale quotients (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) of a denominatored algebra (J, F), we begin with their direct sum Q_{1} + Q_{2} with f : J ⟶ Q_{1} + Q_{2} defined by f(x) = (τ_{1}(x), τ_{2}(x)) for any x ∈ J. The map f is clearly an algebra monomorphism, so that we can take the Martindale quotient (Q̃(f, F), f̃) as in (4.2)(iii) (for Q = Q_{1} + Q_{2}). Moreover, (2.6)(i) implies that Ann(f, F) = 0, so that by (4.2) (Q̃(f, F), f̃) = (Q(f, F), f̃) and we obtain a Martindale quotient

(1) (Q_{1} ∧ Q_{2}, τ_{1} ∧ τ_{2}) := (Q(f, F), f̃) (f̃ the co-restriction of f),

where Q_{1} ∧ Q_{2} consists of all (q_{1}, q_{2}) ∈ Q_{1} + Q_{2} satisfying

(2) there exists I ∈ F such that for any a ∈ I there are x = x(a), y = y(a), z = z(a) ∈ J with

U_{q_{i}}τ_{i}(a) = τ_{i}(x), U_{τ_{i}(a)}q_{i} = τ_{i}(y), q_{i} ∘ τ_{i}(a) = τ_{i}(z) for i = 1, 2.

5.8 Theorem. Martindale quotients Mq(J, F) are directed downwards: (Q_{1} ∧ Q_{2}, τ_{1} ∧ τ_{2}) is the infimum of (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) in Mq(J, F), equivalently, [Q_{1} ∧ Q_{2}, τ_{1} ∧ τ_{2}]_{X} is the infimum of [Q_{1}, τ_{1}]_{X}, [Q_{2}, τ_{2}]_{X} in the poset (Iso_{X}(J, F), ≤).

Proof: Clearly, the restriction of the natural projection π_{i} : Q(f, F) ⟶ Q_{i} (π_{i}((q_{1}, q_{2})) = q_{i}) is an algebra homomorphism such that π_{i}f̃ = τ_{i}, for i = 1, 2, hence (Q(f, F), f̃) ≤ (Q_{i}, τ_{i}) for i = 1, 2. If another (Q, τ) ∈ Mq(J, F) satisfies (Q, τ) ≤ (Q_{i}, τ_{i}) for i = 1, 2, then there exist algebra homomorphisms g_{i} : Q ⟶ Q_{i} such that g_{i}τ = τ_{i}, i = 1, 2. Again we can define the algebra homomorphism g : Q ⟶ Q_{1} + Q_{2} given by g(q) = (g_{1}(q), g_{2}(q)), and it can be readily seen from the construction that g(Q) ⊆ Q(f, F): g(q) satisfies (5.7)(2) for any I ∈ F with I ⊆ I_{1} ∩ I_{2}, where I_{i} ∈ F (i = 1, 2) satisfies β_{τ_{i}(I_{i})}g_{i}(q) ⊆ τ_{i}(J); this is due to the fact that g_{i}τ = τ_{i}, and injectivity of g_{i} (2.4) and τ. Thus we can restrict g in the image to the algebra homomorphism g̃ : Q ⟶ Q(f, F) clearly satisfying g̃τ = f̃, which proves (Q, τ) ≤ (Q(f, F), f̃).

The infimum of (Q_{1}, τ_{1}) ∧ (Q_{2}, τ_{2}) := (Q_{1} ∧ Q_{2}, τ_{1} ∧ τ_{2}) ∈ Mq(J, F) found in (5.8), together with its explicit construction (5.7), and (2.7), gives us another way to describe the order relation ≤.

5.9 Corollary. Given (Q_{1}, τ_{1}), (Q_{2}, τ_{2}) ∈ Mq(J, F), the following are equivalent:

(i) (Q_{1}, τ_{1}) ≤ (Q_{2}, τ_{2}),
(ii) (Q_{1}, τ_{1}) ≅ (Q_{1}, τ_{1}) ∧ (Q_{2}, τ_{2}),

(iii) for any q_{1} ∈ Q_{1}, there exists q_{2} ∈ Q_{2} such that (q_{1}, q_{2}) satisfies (5.7)(2).

Proof: (i) ⟺ (ii): Apply (5.3), (5.4)(1) and (5.8).

(i) ⟹ (iii): If (Q_{1}, τ_{1}) ≤ (Q_{2}, τ_{2}), (Q_{1}, τ_{1}) can play the role of (Q, τ) in the proof of (5.8). Hence g_{1} = Id_{Q_{1}} by (2.7), and, for any q_{1} ∈ Q_{1}, we can take q_{2} := g_{2}(q_{1}).

(iii) ⟹ (ii): As shown in the proof of (5.8), the restriction of the projection π_{1} : Q_{1} ∧ Q_{2} ⟶ Q_{1} is an algebra homomorphism satisfying π_{1}(τ_{1} ∧ τ_{2}) = τ_{1}, hence it is injective by (2.4). Given any q_{1} ∈ Q_{1}, the element q_{2} ∈ Q_{2} as in (iii) satisfies (q_{1}, q_{2}) ∈ Q_{1} ∧ Q_{2}, and, obviously, π_{1}((q_{1}, q_{2})) = q_{1}, which shows that π_{1} : Q_{1} ∧ Q_{2} ⟶ Q_{1} is surjective.

6. Unital Hulls and Martindale Quotients

6.1 Given a Jordan algebra J, a unital hull J_{1} of J is usually understood to be any unital Jordan algebra such that J is a subalgebra of J_{1} and J_{1} = Φ1 + J is generated (as an algebra, equivalently, as a Φ-module) by J and the unit element 1.

More generally, we define a unital hull of a Jordan algebra J to be a pair (J_{1}, μ_{1}), where J_{1} is a unital algebra, and μ_{1} : J ⟶ J_{1} is an algebra monomorphism such that J_{1} is generated by μ_{1}(J) and 1_{J_{1}}. A unital hull (J_{1}, μ_{1}) of J will be said tight over J if every nonzero ideal of J_{1} hits μ_{1}(J), i.e., I ∩ μ_{1}(J) ≠ 0 for any nonzero ideal I of J_{1}.

6.2 Proposition. Given a Jordan algebra J with Zann_{J}(J) = 0, all tight unital hulls are isomorphic to (J̌, μ̌) := (π(Ĵ), πι), for Ĵ the free unital hull of J, ι : J ⟶ Ĵ the natural inclusion, and π : Ĵ ⟶ Ĵ/ Zann_{J}(ι(J)) the natural projection. If (J_{1}, μ_{1}) is a tight unital hull of J, then there is a unique isomorphism f : J̌ ⟶ J_{1} such that f μ̌ = μ_{1}.

Proof: Given a tight unital hull (J_{1}, μ_{1}) of J, the free unital hull Ĵ = Φ1 ⊕ J has a canonical unital algebra epimorphism g : Ĵ ⟶ J_{1} such that gι = μ_{1}. By (0.7) the hypothesis Zann_{J}(J) = 0 guarantees that Zann_{J}(ι(J)) is the maximum ideal of Ĵ not hitting ι(J). But Ker g is also an ideal of Ĵ not hitting ι(J) [since gι = μ_{1} is a monomorphism], and is maximal with respect to this property. Indeed, if I ⊃ Ker g were a bigger ideal then g(I) would be a nonzero ideal of J_{1}, so by tightness of J_{1} over J we would have μ_{1}(J) ∩ g(I) ≠ 0: there are nonzero z ∈ I, a ∈ J with 0 ≠ g(z) = μ_{1}(a) = g(ι(a)). Then z − ι(a) ∈ Ker g ⊆ I and ι(a) = z − (z − ι(a)) ∈ I, so 0 ≠ ι(a) ∈ I ∩ ι(J), and the bigger ideal I would hit ι(J). Thus we must have

Ker g = Zann_{J}(ι(J))

and the isomorphism f : Ĵ/ Ker g = J̌ ⟶ J_{1} induced by g satisfies f μ̌ = fπι = gι = μ_{1}. The uniqueness of f comes from the fact that f is determined on μ̌(J) and ˇ 1.

6.3 Remark: In the above proof, assume, to simplify notation, that J ⊆ Ĵ. Since J ∩Zann_{J}(J) = Zann_{J}(J), Zann_{J}(J) avoids hitting J if and only if Zann_{J}(J) = 0. At the opposite extreme, if J^{2} = U_{J}J = 0 then Zann_{J}(J) = Φ_{0}1 + J ⊇ J for Φ_{0} = {α ∈ Φ | 2αJ = α^{2}J = 0} which is an ideal of Φ. Any 2-dual number ε (2ε = ε^{2} = 0) belongs to Φ_{0}.

Notice that the above elementary approach to tight unital hulls resembles formally that to Martindale quotients of the previous sections. This interaction goes further than mere formal similarity, as shown in the next result.

6.4 Proposition. Let (J, F) be a denominatored algebra, and (J_{1}, μ_{1}) be a tight unital hull of J. Then:

(i) (J_{1}, μ_{1}) is a Martindale quotient of (J, F).

(ii) μ_{1}(F) := {μ_{1}(I) | I ∈ F} is a denominator filter for J_{1}, and, for any Martindale quotient (Q, τ) of (J_{1}, μ_{1}(F)), (Q, τμ_{1}) is a Martindale quotient of (J, F).

(iii) Given any Martindale quotient (Q, τ) of (J, F), and tight unital hull (Q_{1}, ν_{1}) of Q, then (Q_{1}, τ_{1}) for τ_{1} := ν_{1}τ is also a Martindale quotient of (J, F) which can be built from a quotient of a tight unital hull of J as in (ii). In particular, maximal Martindale quotients of (J, F) are always unital and can be viewed as Martindale quotients of tight unital hulls of J.

Proof: We may assume, without loss of generality, that J is contained in J_{1} and μ_{1} is the inclusion map. It is clear that any I ∈ F is an ideal of J_{1}. Moreover, I is sturdy in J_{1} since Zann_{J_{1}}(I) ∩ J = Zann_{J}(I) = 0, which implies Zann_{J_{1}}(I) = 0 by tightness of J_{1} over J. This shows that (J_{1}, F) is a denominatored algebra.

(i) This follows from the Sturdy Ideal Example (3.2).

(ii) The fact that any Martindale quotient (Q, τ) of (J_{1}, F) is a Martindale quotient of (J, F) is straightforward [by (4.1)(i) β_{τ(I)}(q) ⊆ τ(J_{1}) ⟹ β_{(τ(I)^{3})^{3}}(q) ⊆ τ(I) ⊆ τ(J)].

(iii) Notice that Q_{1} = Q(τ_{1}, F) as in (4.2), since the latter clearly contains ν_{1}(Q) and 1_{Q_{1}}. On the other hand Ann(τ_{1}, F) ∩ ν_{1}(Q) = ν_{1}(Ann(τ, F)) = ν_{1}(0) = 0, hence Ann(τ_{1}, F) = 0 by tightness, so that (Q_{1}, τ_{1}) is also a Martindale quotient of (J, F) by (4.2)(iii). Let J_{1} be the subalgebra of Q_{1} generated by τ_{1}(J) and 1_{Q_{1}}, and let σ : J_{1} ⟶ Q_{1} be the inclusion map. Considering the correstriction τ_{1} : J ⟶ J_{1}, we claim that J_{1} is tight over J. Indeed, an ideal L of J_{1} not hitting τ_{1}(J) consists necessarily of elements q ∈ J_{1} ⊆ Q_{1} such that β_{τ_{1}(J)}(q) ⊆ L ∩ τ_{1}(J) = 0, which are zero by (2.6)(i) since (Q_{1}, τ_{1}) is a Martindale quotient of (J, F). Finally, it is obvious that (Q_{1}, σ) is a Martindale quotient of (J_{1}, F_{1}), where F_{1} = {τ_{1}(I) | I ∈ F} is a denominator filter of J_{1} by tightness of J_{1} over J.

6.5 Remark: We can obtain an alternative proof of (3.3) by using (6.4): Let (Q_{1}, ν_{1}) be a tight unital hull of Q. By (6.4)(iii), (Q_{1}, τ_{1}) for τ_{1} = ν_{1}τ is another Martindale quotient of (J, F). But it is easy to show that q = τ_{1}(1_{J}) − 1_{Q_{1}} satisfies β_{τ_{1}(J)}(q) ⊆ U_{q}τ_{1}(J) + U_{τ_{1}(J)}q + q ∘ τ_{1}(J) = 0, hence q = 0 by (2.6)(i), i.e., 1_{Q_{1}} = τ_{1}(1_{J}) ∈ ν_{1}(Q). Thus Q_{1} = ν_{1}(Q), ν_{1} is an isomorphism, and Q is unital with unit ν_{1}^{−1}(1_{Q_{1}}) = τ(1_{J}).

Acknowledgments

This paper originated during a visit of the first named author to the Department of Mathematics of the University of Virginia. He thanks the Department and his Faculty members for their support and hospitality. Both authors want to thank Professors John R. Faulkner and Teresa Cortés for their insightful and valuable comments.

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