Herstein’s Theorems and Simplicity of Hermitian Jordan Systems ^{1}

José A. Anquela, Teresa Cortés, Esther García ^{2}

Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Abstract: In this paper we extend Herstein’s first construction relating associative and Jordan ideals to pairs and triple systems. As a consequence we show that an associative pair or triple systems is simple if and only if its Jordan symmetrization is simple. We also generalize Herstein’s second construction to ample subsystems of associative algebras, pairs and triple systems, which provides information on their simplicity when the associative structure is simple.

Introduction

In [10, 11, 12] Herstein gave constructions associating with any ideal in a Jordan algebra of type R^{(+)} (resp. H(R, ∗) in characteristic≠ 2) an ideal (resp. a ∗-ideal) of the associative algebra (resp. algebra with involution) R. McCrimmon in [15] extends the results of Herstein to algebras over arbitrary rings of scalars (without restrictions on the characteristic) and use them to relate the radicals of R^{(+)} and H(R, ∗) with the radical of R.

A glance to the classification theorems of strongly prime and simple Jordan algebras [18, 15.2, 15.5] gives us an idea of the importance of having information about Jordan algebras of types R^{(+)} and H_{0}(R, ∗): Simple Jordan algebras of hermitian type are essentially of these forms, for a simple associative algebra R, while strongly prime hermitian Jordan algebras are given in terms of that type of algebras, for a prime associative algebra R.

This paper is aimed to extending to associative pairs and triple systems the constructions given in [15], in order to characterize the simplicity of simple Jordan pairs and triple systems of hermitian type [1, 4.4, 5.5].

The first section is devoted to the so-called First Herstein Construction relating the ideals of R and R^{(+)} when R is an associative pair or triple system. The explicit construction of ideals of R out of ideals of R^{(+)} is derived from [15, The First Herstein Construction] by using homotopes and local algebras. As a consequence, we show that Jordan systems of type R^{(+)} are simple if and only if the corresponding associative systems R are simple.

1 Partially supported by the DGES, PB97-1069-C02-02.

2 Supported by an F.P.I. Grant (Ministerio de Educación y Cultura).

When one removes restrictions on the characteristic in the classification results of Jordan systems, ample subsystems H_{0}(R, ∗) replace the whole systems of symmetric elements H(R, ∗). Thus, the inheritance of regularity to H_{0}(R, ∗) from regularity of R in the case of algebras (see, for example, the proof of [8, 4.9]) always require the consideration of core ideals introduced in [15]. Since core ideals are just ideals in their ample hulls, which are ample subspaces, a natural thing to do is trying to extend McCrimmon’s results of [15], dealing with ideals of H(R, ∗), to the more general setting of ideals of an ample subspace H_{0}(R, ∗). This is done in the second section and generalized to pairs and triple systems in the third section of the paper. As a consequence, we obtain information on the simplicity of Jordan systems which are ample subsystems of the form H_{0}(R, ∗) for an associative system R with involution.

Finally, in the fourth section, we give some consequences of the previous constructions. On the one hand, the information on the simplicity of Jordan systems of hermitian type can be used to obtain converses for the classifications of simple Jordan algebras [18, 15.5] and i-special simple Jordan pairs and triple systems [1, 4.4, 5.5; 3, 8.14, 9.6]. On the other hand, the First Herstein Construction for pairs and triple systems provides information on the heart of associative algebras having simple local algebras, i.e., completing the results on local-to-global inheritance of simplicity for associative systems given in [5].

0. Preliminaries

0.1 We will deal with associative and Jordan algebras, pairs and triple systems over an arbitrary ring of scalars Φ. The reader is referred to [1, 13, 14, 18] for basic results, notation and terminology, though we will stress some definitions and basic notions.

—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 x∘y, U_{x,z}y = {x, y, z} = V_{x,y}z, respectively.

—For a Jordan pair V = (V^{+}, V^{−}) we will denote the products by Q_{x}y, for any x ∈ V^{σ}, y ∈ V^{−σ}, σ = ±, with linearizations denoted by Q_{x,z}y = {x, y, z} = D_{x,y}z.

—A Jordan triple system T is given by its products P_{x}y, for any x, y ∈ T, with linearizations denoted by P_{x,z}y = {x, y, z} = L_{x,y}z.

0.2 One can obtain Jordan systems from associative systems by symmetrization: If R is an associative algebra, we can obtain a Jordan algebra denoted by R^{(+)}, over the same Φ-module, with products built out of the associative product by x^{2} = xx, U_{x}y = xyx, for any x, y ∈ R. Similarly, a Jordan pair (resp. Jordan triple system) R^{(+)} can be obtained from an associative pair R = (R^{+}, R^{−}) (resp. an associative triple system R) by defining Q_{x}y = xyx, for any x ∈ R^{σ}, y ∈ R^{−σ}, σ = ± (resp. P_{x}y = xyx, for any x, y ∈ R).

A Jordan system (algebra, pair or triple system) is said to be special if it is a subsystem of R^{(+)} for some associative system R.

0.3 A particularly important example of special Jordan systems are ample subspaces or subpairs of associative systems with involution:

—If R is an associative algebra with involution ∗, a Φ-submodule H_{0}(R, ∗) contained in the set of symmetric elements H(R, ∗) is said to be an ample subspace of R if it contains all traces and norms of the elements of R ({x} = x + x^{∗}, xx^{∗} ∈ H_{0}(R, ∗) for any x ∈ R) and xH_{0}(R, ∗)x^{∗} ⊆ H_{0}(R, ∗) for any x ∈ R [15, p. 387; 18, 0.8′].

—If R = (R^{+}, R^{−}) is an associative pair with polarized involution ∗, an ample subpair H_{0}(R, ∗) = (H_{0}^{+}, H_{0}^{−}) is a pair of submodules of symmetric elements (H_{0}^{σ} ⊆ H(R^{σ}, ∗)) containing all traces {x} = x + x^{∗} of the elements of R^{σ} and satisfying xH_{0}^{−σ}x^{∗} ⊆ H_{0}^{σ}, for any x ∈ R^{σ}, σ = ±. To obtain the notion of ample subsystem of an associative triple system R with involution, simply forget the superscripts in the case of pairs [1, p. 209-210; 6, 1.7].

Clearly, H_{0}(R, ∗) is a subsystem of R^{(+)}.

When dealing with pairs, involutions will be assumed to be of polarized type.

0.4 A Jordan algebra gives rise to a Jordan triple system by simply forgetting the squaring and letting P = U. By doubling any Jordan triple system T one obtains the double Jordan pair V (T) = (T, T) with products Q_{x}y = P_{x}y, for any x, y ∈ T. From a Jordan pair V = (V^{+}, V^{−}) one can get a (polarized) Jordan triple system T (V) = V^{+} ⊕ V^{−} by defining P_{x^{+}⊕x^{−}}(y^{+} ⊕ y^{−}) = Q_{x^{+}}y^{−} ⊕ Q_{x^{−}}y^{+} [14, 1.13, 1.14]

Similarly, one can consider the underlying triple system of an associative algebra, as well as functors V ( ) and T ( ) between the categories of associative pairs and associative triple systems.

0.5 Regularity conditions through the functors V and T [5, 0.3; 1, p. 230; 6, 1.12]:

(i) Let T be a Jordan (resp. associative) triple system.

(a) T is nondegenerate (resp. semiprime) if and only if V (T) is nondegenerate (resp. semiprime), (b) If V (T) is strongly prime (resp. prime), then T is strongly prime (resp.

prime),

(c) If V (T) is simple, then T is simple.

(ii) Let V be a Jordan (resp. associative) pair.

(a) V is nondegenerate (resp. semiprime) if and only if T (V) is nondegenerate (resp. semiprime),

(b) V is strongly prime (resp. prime) if and only if T (V) is strongly prime (resp. prime),

(c) V is simple if and only if T (V) is simple.

(iii) Let J be a Jordan (resp. associative) algebra.

(a) J is nondegenerate (resp. semiprime) if and only if V (J) is nondegenerate (resp. semiprime),

(b) J is strongly prime (resp. prime) if and only if V (J) is strongly prime (resp. prime).

If R is an associative system with involution, primeness can be replaced by ∗- primeness in (i)(b) (with a straightforward proof) and (ii)(b) [4, 1.4(i)].

0.6 Local algebras of Jordan and associative systems, introduced in [19], are one of the most useful tools to study primitivity of Jordan pairs and triple systems. Together with homotopes, they are the way to connect the categories of algebras and pairs and triple systems [2]:

—Given an associative pair R = (R^{+}, R^{−}) and a ∈ R^{−σ}, the Φ-module R^{σ} becomes an associative algebra, denoted R^{σ(a)} and called the a-homotope of R, with product

x ·_{a} y = xay,

for any x, y ∈ R^{σ}. The set

Ker_{R} a = Ker a = {x ∈ R^{σ} | axa = 0}

is an ideal of R^{σ(a)} and the quotient

R_{a}^{σ} = R^{σ(a)}/ Ker a

is an associative algebra called the local algebra of R at a.

—Given a Jordan pair V = (V^{+}, V^{−}) and a ∈ V^{−σ}, the Φ-module V^{σ} becomes a Jordan algebra, denoted V^{σ(a)} and called the a-homotope of V, with products

x^{(2,a)} = Q_{x}a, U_{x}^{(a)}y = Q_{x}Q_{a}y

for any x, y ∈ V^{σ}. The set

Ker_{V} a = Ker a = {x ∈ V^{σ} | Q_{a}x = Q_{a}Q_{x}a = 0}

is an ideal of V^{σ(a)} and the quotient

V_{a}^{σ} = V^{σ(a)}/ Ker a

is a Jordan algebra called the local algebra of V at a. When V is nondegenerate or special, Ker_{V} a = {x ∈ V^{σ} | Q_{a}x = 0}.

—Homotopes and local algebras of a Jordan or associative algebra or triple system J at an element a are simply those of the pair V (J).

The above notions are compatible with the functors V ( ) and T ( ) as well as with symmetrizations and ample subspaces (cf. [4, 0.5]).

0.7 Recall some basic facts concerning semiprimeness and nondegeneracy: It is well-known that an associative algebra R is semiprime if and only if it is nondegenerate (i.e., R^{(+)} is nondegenerate). D’Amour in [1, 1.18] shows the corresponding fact for triple systems, which can be easily extended to pairs by using (0.5)(ii)(a). Thus:

(i) An associative algebra, pair or triple system R is semiprime if and only R^{(+)} is nondegenerate.

In the proof of [4, 4.6(i)] it is shown that ample subpairs of semiprime associative pairs are nondegenerate. Using (0.5)(i)(a) and (0.5)(iii)(a) the same holds for triple systems and algebras:

(ii) If R is a semiprime associative algebra, pair or triple system with involution and H_{0}(R, ∗) is an ample subspace, subpair or subsystem of symmetric elements of R, then H_{0}(R, ∗) is nondegenerate. Moreover, in the case of algebras, H_{0}(R, ∗) is automatically nonzero whenever R is semiprime and nonzero.

(If 0 ≠ R is a semiprime algebra and H_{0}(R, ∗) = 0, then x + x^{∗} = 0 for any x ∈ R, hence x^{∗} = −x for any x ∈ R, and x^{2} = xx = −xx^{∗} = 0, which implies that R is anticommutative. Now, for any x, y ∈ R, xyx = −xxy = 0, i.e., xRx = 0 for any x ∈ R, which is impossible since R^{(+)} is nondegenerate by (i).)

Finally we also remark the well-known fact that ideals inherit nondegeneracy:

(iii) If J is a nondegenerate Jordan algebra, pair or triple system and I is a nonzero ideal of J, then I is nondegenerate.

0.8 Local inheritance of nondegeneracy. Local algebras inherit regularity conditions from Jordan pairs (cf. [2, 6]). We will use the following inheritance of nondegeneracy [2, 2.1.3]: If V is a nondegenerate Jordan pair, then every local algebra V_{b}^{σ} (0 ≠ b ∈ V^{−σ}, σ = ±) is nondegenerate.

1. The First Herstein Construction

1.1 First Herstein Construction for Algebras: In [15, p. 384], Mc-Crimmon shows that for an associative algebra R, ideals of R and R^{(+)} are deeply related. Indeed:

(i) Let R be an associative algebra and I be an ideal of R^{(+)}. Then, for any b ∈ I, the principal ideal L = Φb^{2} + Rb^{2} + b^{2}R + Rb^{2}R of R generated by b^{2} is contained in I.

Moreover, if R is semiprime and 0 ≠ I then there necessarily exists some b ∈ I such that b^{2} ≠ 0, so that:

(ii) If R is a semiprime associative algebra, then every nonzero ideal of R^{(+)} contains a nonzero ideal of R.

1.2 As a consequence of (1.1) McCrimmon obtains in [15, Th. 4]:

(i) An associative algebra R is simple if and only if R^{(+)} is a simple Jordan algebra.

The same arguments together with (0.7)(i), yield:

(ii) An associative algebra R is prime if and only if R^{(+)} is a strongly prime Jordan algebra.

In this section we will apply (1.1) to homotopes of associative pairs to obtain pair and triple system analogues of the above results.

1.3 Lemma. Let R be a semiprime associative pair, and I be an ideal of R^{(+)}. Then, for any 0 ≠ b ∈ I^{σ}, Q_{b}Q_{I^{−σ}}b ≠ 0.

Proof: Let 0 ≠ b ∈ I^{+} satisfy

Q_{b}Q_{I^{−}}b = 0.  (1)

Since I is an ideal of R^{(+)}, I^{−} is an ideal of (R^{(+)})^{−(b)} = (R^{−(b)})^{(+)}, hence Ĩ = (I^{−} + Ker b)/ Ker b is an ideal of (R^{(+)})_{b}^{−} = (R_{b}^{−})^{(+)}. Moreover Ĩ^{3} = 0. Indeed, for any x̃ = x + Ker b, ỹ = y + Ker b ∈ Ĩ (x, y ∈ I^{−}),

Ux̃ỹ = Q_{x}Q_{b}y + Ker b = xbybx + Ker b = 0

since xbybx ∈ Ker b: b(xbybx)b = b{x, b, y}bxb−bybxbxb ∈ b(Q_{I^{−}}b)bxb+byb(Q_{I^{−}}b)b = (Q_{b}Q_{I^{−}}b)xb + by(Q_{b}Q_{I^{−}}b) = 0 by (1).

By (0.7)(i), R^{(+)} is nondegenerate. Thus (R^{(+)})_{b}^{−} is nondegenerate by (0.8), hence semiprime, and Ĩ = 0, i.e., I^{−} ⊆ Ker b, i.e., Q_{b}I^{−} = 0, which is impossible since I is nondegenerate by (0.7)(iii).

By (+, −)-symmetry, there does not exist any 0 ≠ b ∈ I^{−} such that Q_{b}Q_{I^{+}}b = 0.

1.4 Proposition. Let R be an associative pair, and I be an ideal of R^{(+)}, then, for any b ∈ I^{−σ}, x ∈ I^{σ}, σ = ±, R^{σ}bxbxbR^{σ} ⊆ I^{σ}.

Proof: Put σ = +, for example. Since I^{+} is an ideal of (R^{(+)})^{+(b)} = (R^{+(b)})^{(+)}, R^{+(b)} ·_{b} x ·_{b} x ·_{b} R^{+(b)} ⊆ I^{+} by (1.1)(i), i.e., R^{+}bxbxbR^{+} ⊆ I^{+}.

1.5 Theorem. If R is a semiprime associative pair then any nonzero ideal of R^{(+)} contains a nonzero ideal of R.

Proof: Let I be a nonzero ideal of R^{(+)}.

By (0.7)(i), R^{(+)} is nondegenerate, hence I is nondegenerate by (0.7)(iii), and both I^{+} ≠ 0 and I^{−} ≠ 0. Let 0 ≠ b ∈ I^{−}. By nondegeneracy of I, Q_{b}I^{+} ≠ 0 and there exists x ∈ I^{+} such that 0 ≠ b̄ = bxb = Q_{b}x ∈ I^{−}. By (1.3), Qb̄Q_{I^{+}}b̄ ≠ 0 and there exists y ∈ I^{+} such that z = b̄yb̄yb̄ = bxbybxbybxb = Qb̄Q_{y}b̄ ≠ 0. We have 0 ≠ z ∈ I^{−}.

Let L be the ideal of R generated by z, i.e.,

L^{+} = R^{+}zR^{+}, L^{−} = Φz + zR^{+}R^{−} + R^{−}R^{+}z + R^{−}R^{+}zR^{+}R^{−}.

Now L^{+} = R^{+}b̄yb̄yb̄R^{+} ⊆ I^{+} by (1.4) applied to b̄ = bxb = Q_{b}x ∈ I^{−} and y ∈ I^{+}. But

z = bxbybxbybxb = bx(Q_{b}y)x(Q_{b}y)xb ∈ R^{−}x(Q_{b}y)x(Q_{b}y)xR^{−},

hence L^{−} ⊆ R^{−}x(Q_{b}y)x(Q_{b}y)xR^{−} which is contained in I^{−} by (1.4) applied to x ∈ I^{+} and Q_{b}y ∈ I^{−}. We have shown L ⊆ I and, moreover, L ≠ 0 since z ≠ 0.

We can readily obtain a triple system analogue of (1.5) by using the functor V ( ):

1.6 Corollary. If R is a semiprime associative triple system then any nonzero ideal of R^{(+)} contains a nonzero ideal of R.

Proof: Let I be a nonzero ideal of R^{(+)}, hence 0 ≠ V (I) is an ideal of V (R^{(+)}) = (V (R))^{(+)}. Since R is semiprime, V (R) is semiprime by (0.5)(i)(a) and, by (1.5), there exists a nonzero ideal L = (L^{+}, L^{−}) of V (R) contained in V (I). Now, L^{+} + L^{−} is a nonzero ideal of R contained in I.

The above results (together with (0.7)(i)) provide alternative proofs of [4, 4.5(i), 5.8(i)]. They readily imply pair and triple system analogues of Theorem 4 of [15] too.

1.7 Corollary. Let R be an associative pair (resp. triple system).

(i) R is prime if and only if R^{(+)} is strongly prime.

(ii) R is simple if and only if R^{(+)} is simple.

2. The Second Herstein Construction for Algebras Revisited

In [15], McCrimmon gives a construction to relate the ideals of the Jordan algebra of symmetric elements H(R, ∗) (for R an associative algebra with involution ∗) and the ∗-ideals of R. We will show that the results of [15] can be adapted to study the ideals of an arbitrary ample subspace of symmetric elements H_{0}(R, ∗) of R. In some cases the proofs are direct translations of those given by McCrimmon in [15] and are included by the sake of completeness.

2.1 Let R be an associative algebra with involution ∗, and H_{0} = H_{0}(R, ∗) be an ample subspace of symmetric elements of R. For any ∗-ideal B of R we define

K(B, H_{0}) = {b + b^{∗} + ∑_{i} λ_{i}b_{i}b_{i}^{∗} + ∑_{j} b_{j}h_{j}b_{j}^{∗} | b, b_{i}, b_{j} ∈ B, h_{j} ∈ H_{0}, λ_{i} ∈ Φ}.

Notice that K(B) defined in [15, Section 4] is just K(B, H), where H = H(R, ∗) is the whole set of symmetric elements of R. As mentioned in [15], if 1/2 ∈ Φ then the only ample subspace of R is H and K(B, H_{0}) = K(B, H) is the set of symmetric elements H(B, ∗) of B.

2.2 Lemma. If R is an associative algebra with involution ∗, H_{0} = H_{0}(R, ∗) is an ample subspace of symmetric elements of R and B is a ∗-ideal of R, then K(B, H_{0}) is an ideal of H_{0} contained in B ∩ H_{0}.

Proof: Put K = K(B, H_{0}). From the definition (2.1), it is clear that K is a Φ-submodule of B ∩ H_{0}. In particular:

k ∈ K ⟹ k = k^{∗} ∈ B.  (1)

Now,

— K^{2} ⊆ K: If k ∈ K, then k^{2} = kk^{∗} ∈ K by (1).

— U_{K}H_{0} ⊆ K: If k ∈ K, and h ∈ H_{0}, then U_{k}h = khk = khk^{∗} ∈ K by (1).

— U_{H_{0}}K ⊆ K: For any b ∈ B, h, h_{1} ∈ H_{0}, we have

U_{h}(b + b^{∗}) = U_{h}b + U_{h}b^{∗} = hbh + hb^{∗}h = (hbh) + (hbh)^{∗} ∈ K since hbh ∈ B.

U_{h}(bb^{∗}) = hbb^{∗}h = (hb)(hb)^{∗} ∈ K since hb ∈ B.

U_{h}(bh_{1}b^{∗}) = hbh_{1}b^{∗}h = (hb)h_{1}(hb)^{∗} ∈ K since hb ∈ B.

— K ∘ H_{0} ⊆ K: If k ∈ K and h ∈ H_{0}, then k ∈ B by (1); hence hk ∈ B and k ∘ h = kh + hk = hk + (hk)^{∗} ∈ K since k = k^{∗} by (1).

Given an associative algebra R, R̂ will denote the usual unitization R̂ = Φ1 ⊕ R of R. Any involution ∗ of R extends uniquely to R̂ by defining 1^{∗} = 1.

2.3 Lemma. Let R be an associative algebra with involution ∗, H_{0} = H_{0}(R, ∗) be an ample subspace of symmetric elements of R and I be an ideal of H_{0}. Then:

(i) (bhb)x + x^{∗}(bhb) ∈ I, for any b ∈ I, h ∈ H_{0}, x ∈ R̂.
(ii) xbcbx^{∗} ∈ I, for any b, c ∈ I, x ∈ R̂.
(iii) xbcby^{∗} + ybcbx^{∗} ∈ I, for any b, c ∈ I, x, y ∈ R̂.

Proof: (i) (bhb)x + x^{∗}(bhb) = bh(bx + x^{∗}b) + (bx + x^{∗}b)hb − b(hx^{∗} + xh)b = {b, h, bx + (bx)^{∗}} − U_{b}(xh + (xh)^{∗}) ∈ {I, H_{0}, H_{0}} + U_{I}H_{0} ⊆ I since I is an ideal of H_{0}.

(ii) xbcbx^{∗} = (xb + (xb)^{∗})c(xb + (xb)^{∗}) − xb(cx + (cx)^{∗})b − b(cx + (cx)^{∗})bx^{∗} + (xbx^{∗})cb + bc(xbx^{∗}) − b(x^{∗}cx)b = U_{xb+(xb)^{∗}}c − [xb(cx + (cx)^{∗})b + b(cx + (cx)^{∗})bx^{∗}] + {b, c, xbx^{∗}} − U_{b}(x^{∗}cx) ∈ U_{H_{0}}I + I + {I, I, H_{0}} + U_{I}H_{0} (by (i)) ⊆ I since I is an ideal of H_{0}.

(iii) is a linearization of (ii).

2.4 Proposition. Let R be an associative algebra with involution ∗, H_{0} = H_{0}(R, ∗) be an ample subspace of symmetric elements of R and I be an ideal of H_{0}. For any b, c ∈ I, the ideal B = R̂bcbR̂ of R generated by bcb (which is a ∗-ideal) satisfies K(B, H_{0}) ⊆ I.

Proof: Notice that K(B, H_{0}) is spanned over Φ by elements of the form xbcby+ (xbcby)^{∗}, (xbcby)(xbcby)^{∗}, (xbcby)h(xbcby)^{∗}, (xbcby)(zbcbw)^{∗} + (zbcbw)(xbcby)^{∗} and (xbcby)h(zbcbw)^{∗} + (zbcbw)h(xbcby)^{∗}, where x, y, z, w ∈ R̂, h ∈ H_{0}. But we have:

— xbcby + (xbcby)^{∗} = xbcby + y^{∗}bcbx^{∗} ∈ I by (2.3)(iii),

— (xbcby)(xbcby)^{∗} = xb(cbyy^{∗}bc)bx^{∗} ∈ I by (2.3)(ii), since the element cbyy^{∗}bc = c(by)(by)^{∗}c = U_{c}((by)(by)^{∗}) ∈ U_{I}H_{0} ⊆ I,

— (xbcby)h(xbcby)^{∗} = xb(cbyhy^{∗}bc)bx^{∗} ∈ I by (2.3)(ii), as above, since cbyhy^{∗}bc = c(by)h(by)^{∗}c = U_{c}((by)h(by)^{∗}) ∈ U_{I}H_{0} ⊆ I, — (xbcby)(zbcbw)^{∗} + (zbcbw)(xbcby)^{∗} = xbcb(yw^{∗}bcbz^{∗}) + (yw^{∗}bcbz^{∗})^{∗}bcbx^{∗} ∈ I by

(2.3)(iii),

— (xbcby)h(zbcbw)^{∗} + (zbcbw)h(xbcby)^{∗} = xbcb(yhw^{∗}bcbz^{∗}) + (yhw^{∗}bcbz^{∗})^{∗}bcbx^{∗} ∈ I by (2.3)(iii),

which proves K(B, H_{0}) ⊆ I.

2.5 Lemma. If R is a semiprime associative algebra with involution, H_{0} = H_{0}(R, ∗) is an ample subspace of symmetric elements of R and B is a nonzero ∗- ideal of R, then K(B, H_{0}) ≠ 0.

Proof: If K(B, H_{0}) = 0, then, for any b ∈ B, 0 = b + b^{∗}, hence b^{∗} = −b and b^{2} = bb = −bb^{∗} = 0, which implies that B is anticommutative. Thus, for any b, c ∈ B, bcb = −bbc = 0, and we have shown bBb = 0, for any b ∈ B, which is impossible since B is nondegenerate by (0.7)(iii) (B is an ideal of R^{(+)} which is nondegenerate by (0.7)(i)).

2.6 Theorem. Let R be a semiprime associative algebra with involution ∗, H_{0} = H_{0}(R, ∗) be an ample subspace of symmetric elements of R, and I be a nonzero ideal of H_{0}. Then there exists a ∗-ideal B of R such that 0 ≠ K(B, H_{0}) ⊆ I.

Proof: By (2.4) and (2.5), we just need to find b, c ∈ I such that bcb ≠ 0. But this latter is a straightforward consequence of the fact that I is nondegenerate by (0.7)(iii), since H_{0} is nondegenerate by (0.7)(ii).

As a consequence of the above results, we give an alternative proof of [7, 1.3], as well as the extension of Theorem 5 of [15] to arbitrary ample subspaces.

2.7 Corollary. Let R be an associative algebra with involution ∗, H_{0} = H_{0}(R, ∗) be an ample subspace of symmetric elements of R.

(i) If R is ∗-prime, then H_{0} is strongly prime.

(ii) If R is ∗-simple, then the heart Heart(H_{0}) of H_{0} is a simple Jordan algebra. Moreover, 0 ≠ Heart(H_{0}) = K(R, H_{0}) = H_{0}^{3} = JAlg_{2}(H_{0}) and the nonzero ideals of H_{0} are exactly all Φ-submodules of H_{0} containing Heart(H_{0}) = K(R, H_{0}). So H_{0} is simple if and only if H_{0} = K(R, H_{0}).

(H_{0}^{3} denotes the Jordan cube of H_{0}, i.e., the span of elements of the form U_{x}y, where x, y ∈ H_{0}, and JAlg_{2}(H_{0}) denotes the span of Jordan algebra monomials in elements of H_{0} of degree greater than or equal to two (cf. [5, 1.5(i)]).)

Proof: (i) If R = 0, there is nothing to prove. Otherwise, by (0.7)(ii), 0 ≠ H_{0} is nondegenerate. Moreover, by (2.6), for any pair of nonzero ideals I, L of H_{0}, there exist nonzero ∗-ideals B, C of R such that K(B, H_{0}) ⊆ I and K(C, H_{0}) ⊆ L. By ∗-primeness of R, 0 ≠ B ∩ C, hence, by (2.5), 0 ≠ K(B ∩ C, H_{0}) ⊆ K(B, H_{0}) ∩ K(C, H_{0}) ⊆ I ∩ L, and we have shown that H_{0} is prime.

(ii) By (2.6), every nonzero ideal of H_{0} contains K(R, H_{0}) since R is the only nonzero ∗-ideal of R, thus K(R, H_{0}) ⊆ Heart(H_{0}). By ∗-simplicity, R is nonzero, hence K(R, H_{0}) is a nonzero ideal of H_{0} by (2.2) and (2.5), and Heart(H_{0}) = K(R, H_{0}).

The cubes of ideals are again ideals (see [16, p. 221]), hence H_{0}^{3} is an ideal of H_{0}. Moreover, H_{0}^{3} ≠ 0 since H_{0} is nondegenerate by (0.7)(ii). Thus Heart(H_{0}) ⊆ H_{0}^{3}. Conversely, for any x, y ∈ H_{0}, U_{x}y = xyx = xyx^{∗} ∈ K(R, H_{0}) = Heart(H_{0}) and H_{0}^{3} ⊆ Heart(H_{0}).

Clearly Heart(H_{0}) = H_{0}^{3} ⊆ JAlg_{2}(H_{0}). But JAlg_{2}(H_{0}) is spanned by elements of the form x^{2}, x∘y, U_{x}y and U_{x,z}y, where x, y, z ∈ H_{0}, all of them lying in K(R, H_{0}) (x^{2} = xx^{∗}, x ∘ y = xy + yx = xy + (xy)^{∗}, U_{x}y = xyx = xyx^{∗}, U_{x,z}y = xyz + zyx = xyz + (xyz)^{∗}), which shows JAlg_{2}(H_{0}) ⊆ K(R, H_{0}) = Heart(H_{0}).

Let us show that Heart(H_{0}) is simple. Put K = K(R, H_{0}) = Heart(H_{0}). It is straightforward that K is an ample subspace of symmetric elements of R, hence Heart(K) = K^{3}. But K^{3} = Heart(H_{0})^{3} = Heart(H_{0}) = K by [5, 4.4] applied to the underlying triple system of H_{0} which is semiprime since it is nondegenerate. We have shown that K = Heart(K), hence K does not have proper ideals and K is simple since it is nonzero and nondegenerate by (0.7)(ii).

Finally, every Φ-submodule containing Heart(H_{0}) is an ideal since it contains JAlg_{2}(H_{0}).

2.8 Remark: Notice that kernel ideals in the sense of [15, p. 386] are a particular case of ideals of an ample subspace. Indeed, if I is a kernel ideal of an associative algebra R with involution ∗, then I is an ideal of H_{0} = K(R, H(R, ∗)) + I which is an ample subspace of symmetric elements of R. In [15, The Second Herstein Construction (i)] it is shown that a kernel ideal I contains K(B, H(R, ∗)) for B = R̂bcbR̂, for any b, c ∈ I, while applying our result (2.4) yields K(B, H_{0}) ⊆ I, which is, in general, weaker since K(B, H_{0}) ⊆ K(B, H(R, ∗)).

A core ideal in the sense of [15, p. 387] is just an ideal in its ample hull, which is an ample subspace H_{0} of symmetric elements of R. McCrimmon shows that, for a core ideal I, the ideal B = R̂bcbR̂, for any b, c ∈ I, satisfies K_{0}(B) ⊆ I, where K_{0}(B) is the span of traces and norms of elements of B. Applying (2.4) yields K(B, H_{0}) ⊆ I which is, in general, stronger since K_{0}(B) ⊆ K(B, H_{0}).

Theorem 5 of [15] is contained in (2.7) since H(R, ∗) is an ample subspace (the biggest one). The only non explicitly mentioned fact is that K_{0}(R) is simple if R is ∗-simple, which can also be obtained from (2.7)(ii): K_{0}(R) is an ample subspace whose heart is K(R, K_{0}(R)) = K_{0}(R) by the definition of K_{0}(R).

3. The Second Herstein Construction for Pairs and Triple Systems

This paragraph is devoted to extending to pairs and triple systems the results of the previous section.

3.1 Let R be an associative pair with involution ∗, and H_{0} = H_{0}(R, ∗) be an ample subpair of symmetric elements of R. For any ∗-ideal B of R, σ = ±, we define

KP (B^{σ}, H_{0}) = {b + b^{∗} + ∑_{i} b_{i}h_{i}b_{i}^{∗} | b, b_{i} ∈ B^{σ}, h_{i} ∈ H_{0}^{−σ}}.

If 1/2 ∈ Φ then the only ample subpair of R is the whole set of symmetric elements H = H(R, ∗), and KP (B^{σ}, H_{0}) = KP (B^{σ}, H) is the set of symmetric elements H(B^{σ}, ∗) of B^{σ}.

3.2 Lemma. If R is an associative pair with involution ∗, H_{0} = H_{0}(R, ∗) is an ample subpair of symmetric elements of R and B is a ∗-ideal of R, then for σ = ±, KP (B^{σ}, H_{0}) is a semiideal of H_{0} contained in B^{σ} ∩ H_{0}^{σ}.

Proof: Put K = KP (B^{σ}, H_{0}). From the definition (3.1), it is clear that K is a Φ-submodule of B^{σ} ∩ H_{0}^{σ}. In particular,

k ∈ K ⟹ k = k^{∗} ∈ B.  (1)

Now,

— Q_{K}H_{0}^{−σ} ⊆ K: For any k ∈ K, h ∈ H_{0}, Q_{k}h = khk = khk^{∗} ∈ K using (1).

— {H_{0}^{σ}, H_{0}^{−σ}, K} ⊆ K: For any k ∈ K, h_{1} ∈ H_{0}^{σ}, h_{2} ∈ H_{0}^{−σ}, {h_{1}, h_{2}, k} = h_{1}h_{2}k + kh_{2}h_{1} = h_{1}h_{2}k + (h_{1}h_{2}k)^{∗} ∈ K because h_{1}h_{2}k ∈ B since k ∈ B by (1).

— Q_{H_{0}^{σ}}Q_{H_{0}^{−σ}}K ⊆ K: For any b ∈ B^{σ}, h, h_{1} ∈ H_{0}^{−σ}, h_{2} ∈ H_{0}^{σ},

(a) Q_{h_{2}}Q_{h_{1}}(b + b^{∗}) = h_{2}h_{1}(b + b^{∗})h_{1}h_{2} = h_{2}h_{1}bh_{1}h_{2} + (h_{2}h_{1}bh_{1}h_{2})^{∗} ∈ K since h_{2}h_{1}bh_{1}h_{2} ∈ B^{σ}.

(b) Q_{h_{2}}Q_{h_{1}}(bhb^{∗}) = h_{2}h_{1}bhb^{∗}h_{1}h_{2} = (h_{2}h_{1}b)h(h_{2}h_{1}b)^{∗} ∈ K since h_{2}h_{1}b ∈ B^{σ}.

3.3 Proposition. Let R be an associative pair with involution ∗, H_{0} = H_{0}(R, ∗) be an ample subpair of symmetric elements of R and I be an ideal of H_{0}. For any σ = ±, a, c ∈ I^{σ}, b ∈ I^{−σ}, the ideal B of R generated by babcbab (which is a ∗-ideal) satisfies KP (B^{σ}, H_{0}) ⊆ I^{σ}.

Proof: Put, for example, σ = +. Now, I^{+} is an ideal of the b-homotope H_{0}^{+(b)} which is an ample subspace of R^{+(b)}. By (2.4), the ideal L of R^{+(b)} generated by a ·_{b} c ·_{b} a = abcba satisfies

K(L, H_{0}^{+(b)}) ⊆ I^{+}.  (1)

Notice that

B^{+} = R^{+}babcbabR^{+} = R^{+} ·_{b} a ·_{b} c ·_{b} a ·_{b} R^{+} ⊆ L.  (2)

Now, for any z ∈ B^{+},

— z + z^{∗} ∈ I^{+}: Use (1) since z + z^{∗} ∈ K(L, H_{0}^{+(b)}) by (2),

— zH_{0}^{−}z^{∗} ⊆ I^{+}: z = ∑_{i=1}^{n} x_{i}babcbaby_{i}, where x_{i}, y_{i} ∈ R^{+}; hence, for any h ∈ H_{0}^{−},

zhz^{∗} = ∑_{i=1}^{n} x_{i}babcbaby_{i}hy_{i}^{∗}babcbabx_{i}^{∗}+

∑_{i<j=1}^{n}(x_{i}babcbaby_{i}hy_{j}^{∗}babcbabx_{j}^{∗} + x_{j}babcbaby_{j}hy_{i}^{∗}babcbabx_{i}^{∗}) =

∑_{i=1}^{n}(x_{i}babcba) ·_{b} (y_{i}hy_{i}^{∗}) ·_{b} (x_{i}babcba)^{∗}+

∑_{i<j=1}^{n}(x_{i}babcbaby_{i}hy_{j}^{∗}babcbabx_{j}^{∗} + (x_{i}babcbaby_{i}hy_{j}^{∗}babcbabx_{j}^{∗})^{∗}) ∈ I^{+}

by (1) since x_{i}babcba = x_{i} ·_{b} a ·_{b} c ·_{b} a ∈ L and x_{i}babcbaby_{i}hy_{j}^{∗}babcbabx_{j}^{∗} ∈

R^{+}babcbabR^{+} = B^{+} ⊆ L by (2),

and we have shown KP (B^{+}, H_{0}) ⊆ I^{+}.

3.4 Remark: Unlike for algebras (cf. (2.5)), we may find semiprime associative pairs R with involution having a nonzero ample subpair H_{0} = H_{0}(R, ∗) and a nonzero ∗-ideal B such that KP (B^{+}, H_{0}) = 0 and KP (B^{−}, H_{0}) = 0: Let Φ an arbitrary field of characteristic not two. Let S be an arbitrary semiprime associative Φ-pair with involution ∗ such that H(S, ∗) ≠ 0. Let R = S + (Φ, Φ) and extend to R the involution of S by (x + λ)^{∗} = x^{∗} − λ (x ∈ S^{σ}, λ ∈ Φ, σ = ±). Now H_{0} := H(R, ∗) = H(S, ∗) + 0 is a nonzero ample subpair of R, B := 0 + (Φ, Φ) is a nonzero ∗-ideal of R and KP (B^{σ}, H_{0}) = 0 since B^{σ} ∩ H_{0}^{σ} = 0 (σ = ±).

We need ∗-primeness to have an analogue for pairs of (2.5).

3.5 Lemma. If R is a ∗-prime associative pair with involution, 0 ≠ H_{0} = H_{0}(R, ∗) is an ample subpair of symmetric elements of R and B is a nonzero ∗-ideal of R, then both KP (B^{+}, H_{0}) ≠ 0 and KP (B^{−}, H_{0}) ≠ 0.

Proof: Assume KP (B^{+}, H_{0}) = 0 and notice that R is semiprime since it is ∗-prime. By nondegeneracy of H_{0} (0.7)(ii), both H_{0}^{+} ≠ 0 and H_{0}^{−} ≠ 0. Moreover, R^{(+)} is nondegenerate by (0.7)(i) and so is any nonzero ideal of R^{(+)} by (0.7)(iii). In particular, B^{(+)} is nondegenerate and both B^{+} ≠ 0 and B^{−} ≠ 0. For any b ∈ B^{+} we have:

b^{∗} = −b  (1)

since b + b^{+} ∈ KP (B^{+}, H_{0}) = 0,

bH_{0}^{−}b = 0  (2)

since, for any h ∈ H_{0}^{−}, using (1) yields bhb = −bhb^{∗} ∈ KP (B^{+}, H_{0}) = 0, and

bR^{−}R^{+}H_{0}^{−}R^{+}R^{−}b = 0  (3)

since, for any x, y ∈ R^{−}, z, t ∈ R^{+}, h ∈ H_{0}^{−}, bxz, tyb ∈ B^{+} implies bxz = −(bxz)^{∗} and tyb = −(tyb)^{∗} by (1), hence bxzhtyb = (bxz)^{∗}h(tyb)^{∗} = z^{∗}x^{∗}b^{∗}hb^{∗}y^{∗}t^{∗} = z^{∗}x^{∗}bhby^{∗}t^{∗} = 0 by (2).

Let 0 ≠ b ∈ B^{+}, let M be the ideal of R generated by b, and let L be the ideal of R generated by H_{0}^{−}. Clearly L is a ∗-ideal of R, while M is a ∗-ideal of R by (1). Moreover, both M and L are nonzero. Hence, by ∗-primeness of R, A = M ∩ L is a nonzero ideal of R. Notice that L^{+} = R^{+}H_{0}^{−}R^{+} and M^{−} = R^{−}bR^{−}, hence

A^{−}A^{+}A^{−} ⊆ M^{−}L^{+}M^{−} = R^{−}bR^{−}R^{+}H_{0}^{−}R^{+}R^{−}bR^{−} = 0

by (3), which is impossible by semiprimeness of R.

Similarly KP (B^{−}, H_{0}) can be shown to be nonzero.

3.6 Theorem. Let R be a semiprime associative pair with involution ∗, 0 ≠ H_{0} = H_{0}(R, ∗) be an ample subpair of symmetric elements of R, and I be a nonzero ideal of H_{0}. Then there exists a nonzero ∗-ideal B of R such that KP (B^{+}, H_{0}) ⊆ I^{+} and KP (B^{−}, H_{0}) ⊆ I^{−}. Moreover, if R is ∗-prime, then KP (B^{+}, H_{0}) ≠ 0 and KP (B^{−}, H_{0}) ≠ 0.

Proof: Since R is semiprime, H_{0} is nondegenerate by (0.7)(ii), hence I is nondegenerate by (0.7)(iii). Thus both I^{+} and I^{−} are nonzero and we can take 0 ≠ b ∈ I^{+}. By nondegeneracy of I, Q_{b}I^{−} ≠ 0, and there exists 0 ≠ x ∈ I^{−} such that Q_{b}x = bxb ≠ 0. Again by nondegeneracy of I, there exists a ∈ I^{−} such that bxbabxb ≠ 0 and, hence, there also exists c ∈ I^{−} such that z = bxbabxbcbxbabxb ≠ 0. Let B be the ideal of R generated by z, which is a nonzero ∗-ideal since z ≠ 0 is symmetric.

By (3.3) applied to bxb ∈ I^{+}, a, c ∈ I^{−}, we have KP (B^{−}, H_{0}) ⊆ I^{−}.

Let N be the ideal of R generated by xbabxbcbxbabx. It is clear that B^{+} = Φz+R^{+}R^{−}z+zR^{−}R^{+}+R^{+}R^{−}zR^{−}R^{+} ⊆ R^{+}xbabxbcbxbabxR^{+} = N^{+}, which implies KP (B^{+}, H_{0}) ⊆ KP (N^{+}, H_{0}). But KP (N^{+}, H_{0}) ⊆ I^{+} by (3.3) applied to x ∈ I^{−}, bab = Q_{b}a, bcb = Q_{b}c ∈ I^{+} and therefore KP (B^{+}, H_{0}) ⊆ I^{+}.

Finally, if R is ∗-prime, then KP (B^{+}, H_{0}) ≠ 0 and KP (B^{+}, H_{0}) ≠ 0 by (3.5) since B ≠ 0.

3.7 Corollary. Let R be an associative pair with involution ∗, 0 ≠ H_{0} = H_{0}(R, ∗) be an ample subpair of symmetric elements of R.

(i) If R is ∗-prime, then H_{0} is strongly prime.

(ii) If R is ∗-simple, then the heart Heart(H_{0}) of H_{0} is a simple Jordan pair. Moreover, 0 ≠ Heart(H_{0}) = (KP (R^{+}, H_{0}), KP (R^{−}, H_{0})) = H_{0}^{3} and the nonzero ideals of H_{0} are exactly all pairs of Φ-submodules of H_{0} containing Heart(H_{0}).

(H_{0}^{3} denotes the Jordan cube of H_{0}, i.e., the pair of spans of elements of the form Q_{x}y, where x ∈ H_{0}^{σ}, y ∈ H_{0}^{−σ}, σ = ±.)

Proof: (i) By (0.7)(ii), 0 ≠ H_{0} is nondegenerate. Moreover, by (3.6), for any pair of nonzero ideals I, L of H_{0}, there exist nonzero ∗-ideals B, C of R such that KP (B^{σ}, H_{0}) ⊆ I^{σ} and KP (C^{σ}, H_{0}) ⊆ L^{σ} (for both σ = ±). By ∗-primeness of R, 0 ≠ B ∩ C, hence, by (3.5), 0 ≠ KP ((B ∩ C)^{σ}, H_{0}) ⊆ KP (B^{σ}, H_{0}) ∩ KP (C^{σ}, H_{0}) ⊆ I^{σ} ∩ L^{σ}, σ = ±, which shows H_{0} is prime.

(ii) By (3.6), every nonzero ideal of H_{0} contains (KP (R^{+}, H_{0}), KP (R^{−}, H_{0})) since R is the only nonzero ∗-ideal of R, thus KP (R^{σ}, H_{0}) ⊆ Heart(H_{0})^{σ}, σ = ±. Moreover, 0 ≠ KP (R^{σ}, H_{0}), σ = ±, by (3.5). We claim that the pair (KP (R^{+}, H_{0}), KP (R^{−}, H_{0})) is an ideal of H_{0}: by (2.2), both KP (R^{+}, H_{0}), KP (R^{−}, H_{0}) are semiideals and, for any x ∈ KP (R^{σ}, H_{0}), y ∈ H_{0}^{−σ}, Q_{y}x = yxy = yxy^{∗} ∈ yH_{0}^{σ}y^{∗} ⊆ KP (R^{−σ}, H_{0}). Thus Heart(H_{0}) = (KP (R^{+}, H_{0}), KP (R^{−}, H_{0})) ≠ 0.

It is clear that H_{0}^{3} is an ideal of H_{0}. Moreover, H_{0}^{3} ≠ 0 since H_{0} is nondegenerate by (0.7)(ii). Thus Heart(H_{0}) ⊆ H_{0}^{3}. Conversely, for any x ∈ H_{0}^{σ}, y ∈ H_{0}^{−σ}, Q_{x}y = xyx = xyx^{∗} ∈ KP (R^{σ}, H_{0}) = Heart(H_{0})^{σ}, and H_{0}^{3} ⊆ Heart(H_{0}).

Let us show that Heart(H_{0}) is simple. Put K = (KP (R^{+}, H_{0}), KP (R^{−}, H_{0})) = Heart(H_{0}). It is straightforward that K is a nonzero ample subpair of symmetric elements of R, hence Heart(K) = K^{3}. But K^{3} = Heart(H_{0})^{3} = Heart(H_{0}) = K by [5, 4.4] applied to the Jordan triple system T (H_{0}) (T (H_{0}) is semiprime since it is nondegenerate by (0.5)(ii)(a), and Heart(T (H_{0})) = T (Heart(H_{0})) ≠ 0 as in the proof of [5, 4.7(i)]). We have shown that K = Heart(K), hence K does not have proper ideals, and K is simple since it is nondegenerate by (0.7)(ii), hence nontrivial.

Finally, every pair of Φ-submodules containing Heart(H_{0}) is an ideal since it contains H_{0}^{3}.

3.8 Remark: As for algebras, given an associative pair R with involution ∗, we can consider the smallest ample subpair KP_{0}(R) = (KP_{0}(R^{+}), KP_{0}(R^{−})), where KP_{0}(R^{σ}) = {r + r^{∗} | r ∈ R^{σ}}, σ = ±. By (3.7)(ii), if R is ∗-simple, then KP_{0}(R) is simple since KP (R^{σ}, KP_{0}(R)) = KP_{0}(R^{σ}) for both σ = ±.

We also have triple versions of the above results, stated, in general, by simply deleting the superscripts ±. However, in some cases the proofs cannot be translated due to the difference between ideals generated in pairs and triple systems. We overcome the difficulties by using the functor V ( ).

3.9 Let R be an associative triple system with involution ∗, and H_{0} = H_{0}(R, ∗) be an ample subsystem of symmetric elements of R. For any ∗-ideal B of R we define

KT (B, H_{0}) = {b + b^{∗} + ∑_{i} b_{i}h_{i}b_{i}^{∗} | b, b_{i} ∈ B, h_{i} ∈ H_{0}}.

If 1/2 ∈ Φ then the only ample subsystem of R is the whole set of symmetric elements H = H(R, ∗), and KT (B, H_{0}) = KT (B, H) is the set of symmetric elements H(B, ∗) of B.

3.10 Lemma. If R is an associative triple system with involution ∗, H_{0} = H_{0}(R, ∗) is an ample subsystem of symmetric elements of R and B is a ∗-ideal of R, then KT (B, H_{0}) is a semiideal of H_{0} contained in B ∩ H_{0}.

Proof: Delete the superscripts ± and replace Q by P in the proof of (3.2).

3.11 Remark: We will not give a triple translation of (3.3) since it is explicitly based on the way ideals are generated in associative pairs. A triple version of (3.3) would then require introducing semiideals, hence the extension of (3.9) to semiideals, which is more cumbersome than illuminating. In case a version of (3.3) is needed for an associative triple system with involution R, then one can consider V (R) and apply (3.3) directly.

3.12 Remark: As for pairs (3.4), we may find semiprime associative triple systems R with involution having a nonzero ample subsystem H_{0} = H_{0}(R, ∗) and a nonzero ∗-ideal B such that KT (B, H_{0}) = 0: Let Φ an arbitrary field of characteristic not two. Let S be an arbitrary semiprime associative Φ-triple system with involution ∗ such that H(S, ∗) ≠ 0. Let R = S + Φ and extend to R the involution of S by (x + λ)^{∗} = x^{∗} − λ (x ∈ S, λ ∈ Φ). Now H_{0} := H(R, ∗) = H(S, ∗) + 0 is a nonzero ample subsystem of R, B := 0 + Φ is a nonzero ∗-ideal of R and KT (B, H_{0}) = 0 since B ∩ H_{0} = 0.

3.13 Lemma. If R is a ∗-prime associative triple system with involution, 0 ≠ H_{0} = H_{0}(R, ∗) is an ample subsystem of symmetric elements of R and B is a nonzero ∗-ideal of R, then KT (B, H_{0}) ≠ 0.

Proof: Due to the way ideals are generated in triple systems, we cannot merely follow the proof of (3.5).

Let us assume KT (B, H_{0}) = 0. By deleting the superscripts ± and replacing KP by KT in the proof of (3.5) we get

bH_{0}b = bRRH_{0}RRb = 0  (1)

for any b ∈ B.

We know that H_{0} is nondegenerate by (0.7)(ii), hence P_{H_{0}}H_{0} ≠ 0. Let A be the ideal of R generated by P_{H_{0}}H_{0}, i.e.,

A = P_{H_{0}}H_{0} + (P_{H_{0}}H_{0})RR + RRP_{H_{0}}H_{0} + R(P_{H_{0}}H_{0})R + RR(P_{H_{0}}H_{0})RR.

Since P_{H_{0}}H_{0} ⊆ H_{0} ∩ H_{0}H_{0}H_{0}, we have A ⊆ H_{0} + RRH_{0}RR, and (1) implies

bAb = 0.  (2)

for any b ∈ B.

Let C = A ∩ B which is a nonzero ∗-ideal of R by ∗-primeness of R. Notice that R^{(+)} is nondegenerate by (0.7)(i) since R is semiprime. Hence C^{(+)} is nondegenerate by (0.7)(iii) and, at the same time, P_{C}C = 0 by (2), which is impossible.

3.14 Theorem. Let R be a semiprime associative triple system with involution ∗, 0 ≠ H_{0} = H_{0}(R, ∗) be an ample subsystem of symmetric elements of R, and I be a nonzero ideal of H_{0}. Then there exists a nonzero ∗-ideal B of R such that KT (B, H_{0}) ⊆ I. Moreover, if R is ∗-prime, then KT (B, H_{0}) ≠ 0.

Proof: By (0.5)(i)(a) V (R) is a semiprime associative pair with involution and V (I) is an ideal of V (H_{0}), which is a nonzero ample subpair of symmetric elements of V (R). Using (3.6), there exists a nonzero ∗-ideal A of V (R) satisfying both KP (A^{+}, V (H_{0})) ⊆ V (I)^{+} = I and KP (A^{−}, V (H_{0})) ⊆ V (I)^{−} = I. Take B = A^{+} + A^{−} which is trivially a nonzero ∗-ideal of R. Now, for any z ∈ B, h ∈ H, put z = x + y, where x ∈ A^{+}, y ∈ A^{−}. Hence

— z + z^{∗} = (x + x^{∗}) + (y + y^{∗}) ∈ KP (A^{+}, V (H_{0})) + KP (A^{−}, V (H_{0})) ⊆ I,

— zhz^{∗} = (x + y)h(x^{∗} + y^{∗}) = xhx^{∗} + yhy^{∗} + xhy^{∗} + yhx^{∗} = xhx^{∗} + yhy^{∗} + xhy^{∗} + (xhy^{∗})^{∗} ∈ KP (A^{+}, V (H_{0})) + KP (A^{−}, V (H_{0})) ⊆ I since xhy^{∗} ∈ A^{+}, and we have shown that KT (B, H_{0}) ⊆ I.

Finally, if R is ∗-prime, then KT (B, H_{0}) ≠ 0 by (3.13) since B ≠ 0.

3.15 Corollary. Let R be an associative triple system with involution ∗, 0 ≠ H_{0} = H_{0}(R, ∗) be an ample subsystem of symmetric elements of R.

(i) If R is ∗-prime, then H_{0} is strongly prime.

(ii) If R is ∗-simple, then the heart Heart(H_{0}) of H_{0} is a simple Jordan triple system. Moreover, 0 ≠ Heart(H_{0}) = KT (R, H_{0}) = H_{0}^{3} and the nonzero ideals of H_{0} are exactly all Φ-submodules of H_{0} containing Heart(H_{0}).

(H_{0}^{3} denotes the Jordan cube of H_{0}, i.e., the span of elements of the form P_{x}y, where x, y ∈ H_{0}.)

Proof: Delete the superscripts ±, replace Q by P and KP by KT, and make obvious changes in the proof of (3.7) .

3.16 Remark: As for pairs (3.8), given an associative triple system R with involution ∗, we can consider the smallest ample subsystem KT_{0}(R) = {r + r^{∗} | r ∈ R}. By (3.15)(ii), if R is ∗-simple, then KT_{0}(R) is simple since KT (R, KT_{0}(R)) = KT_{0}(R).

4. Consequences and Further Questions

The results concerning the simplicity of Jordan systems obtained from associative systems [15, Th. 4], (1.7)(ii), (2.7)(ii), (3.7)(ii) and (3.15)(ii) show the simplicity of Jordan systems of hermitian type in the sense of [1, 4.4, 5.5]. Therefore, they provide converses of the descriptions of simple Jordan algebras [18, 15.5] and i-special simple Jordan pairs and triple systems [1, 4.4, 5.5; 3, 8.14, 9.6]:

4.1 Theorem. A Jordan algebra is simple if and only if one of the following holds:

(i) it is isomorphic to R^{(+)} for a simple associative algebra R;

(ii) it is isomorphic to H_{0}^{3}, where H_{0} = H_{0}(R, ∗) is an ample subspace of symmetric elements of a simple associative algebra R with involution;

(iii) it is isomorphic to algebras of types (II) or (III) of [18, 15.5].

Proof: By [18, 15.5], a simple Jordan algebra satisfies (i), (iii) or it is isomorphic to an ample subspace H_{0} = H_{0}(R, ∗) of symmetric elements of a simple associative algebra R with involution. Since H_{0} is simple and H_{0}^{3} is a nonzero ideal of H_{0}, H_{0}^{3} = H_{0}. Conversely, it is known that all algebras of types (II) and (III) of [18, 15.5] are simple (see the comments after [18, 15.5]); algebras satisfying (i) are simple by [15, Th. 4]; algebras satisfying (ii) are simple by (2.7)(ii).

4.2 Theorem. A Jordan pair is i-special and simple if and only if one of the following holds:

(i) it is isomorphic to R^{(+)} for a simple associative pair R;

(ii) it is isomorphic to H_{0}^{3}, where H_{0} = H_{0}(R, ∗) is an ample subpair of symmetric elements of a simple associative pair R with involution;

(iii) it is isomorphic to one of the Jordan pairs listed in [3, 8.14].

Proof: By [1, 5.5], an i-special simple Jordan pair either satisfies a Clifford homotope PI and, hence, it is isomorphic to one of the Jordan pairs listed in [3, 8.14], or it satisfies (i), or it is isomorphic to an ample subpair H_{0} = H_{0}(R, ∗) of symmetric elements of a simple associative pair R with involution. Since H_{0} is simple and H_{0}^{3} is a nonzero ideal of H_{0}, H_{0}^{3} = H_{0}. Conversely, it is clear that all Jordan pairs satisfying (i), (ii) or (iii) are special, hence they are i-special. Those pairs satisfying (iii) are simple since, as mentioned in [3], they are included in Loos’ Classical List [14, 12.12]; Jordan pairs satisfying (i) are simple by (1.7)(ii); Jordan pairs satisfying (ii) are simple by (3.7)(ii).

4.3 Theorem. A Jordan triple system is i-special and simple if and only if one of the following holds:

(i) it is isomorphic to R^{(+)} for a simple associative triple system R;

(ii) it is isomorphic to H_{0}^{3}, where H_{0} = H_{0}(R, ∗) is an ample subsystem of symmetric elements of a simple associative triple system R with involution;

(iii) it is isomorphic to one of the Jordan triple systems listed in [3, 9.6].

Proof: By [1, 4.4], an i-special simple Jordan triple system either satisfies a Clifford homotope PI and, hence, it is isomorphic to one of the Jordan pairs listed in [3, 9.6], or it satisfies (i), or it is isomorphic to an ample subsystem H_{0} = H_{0}(R, ∗) of symmetric elements of a simple associative triple system R with involution. Since H_{0} is simple and H_{0}^{3} is a nonzero ideal of H_{0}, H_{0}^{3} = H_{0}. Conversely, it is clear that all Jordan triple systems satisfying (i), (ii) or (iii) are special, hence they are i-special. Those triple systems satisfying (iii) are simple by [3, 9.6]; Jordan triple systems satisfying (i) are simple by (1.7)(ii); Jordan triple systems satisfying (ii) are simple by (3.15)(ii).

4.4 Remark: (i) All Jordan systems appearing in (4.2) and (4.3) are indeed special, hence those results can be viewed as a description of special simple systems.

(ii) The problem of whether there exists a simple associative algebra, pair or triple system R with involution having a nonzero ample subsystem H_{0} = H_{0}(R, ∗) which is not simple (i.e., H_{0} ≠ H_{0}^{3}) is still open.

We will next use the First Herstein Construction to improve the results on localto-global inheritance of simplicity for associative systems given in [5].

4.5 Theorem. Let R be an associative triple system (resp. algebra) such that for any 0 ≠ a ∈ R, R_{a} is a simple algebra. Then Heart R = P_{R}R (resp. Heart R = U_{R}R) is a simple associative triple system (resp. algebra).

Proof: By [5, 4.6(ii)], we just need to show that Heart R ⊆ P_{R}R (resp. Heart R ⊆ U_{R}R): R is semiprime since Heart R ≠ 0 is simple, hence R^{(+)} is nondegenerate by (0.7)(i), and P_{R}R (resp. U_{R}R) is a nonzero ideal of R^{(+)}. By (1.6) (resp. (1.1)(ii)), there exists a nonzero ideal B of R contained in P_{R}R (resp. U_{R}R), hence Heart R ⊆ P_{R}R (resp. Heart R ⊆ U_{R}R) since Heart R ⊆ B.

4.6 Theorem. Let R be a semiprime associative pair such that for some σ = + or σ = −, R_{a}^{σ} is a simple algebra for any 0 ≠ a ∈ R^{−σ}. Then H := Heart R is a simple associative pair given by H^{σ} = R^{σ}(Q_{R^{−σ}}R^{σ})R^{σ} = Q_{R^{σ}}Q_{R^{−σ}}R^{σ}, H^{−σ} = Q_{R^{−σ}}R^{σ}.

Proof: Put σ = +, for example.

By [5, 4.7(ii)] and the obvious fact Q_{R^{+}}Q_{R^{−}}R^{+} ⊆ R^{+}(Q_{R^{−}}R^{+})R^{+} we just need to show H ⊆ (Q_{R^{+}}Q_{R^{−}}R^{+}, Q_{R^{−}}R^{+}). Since R is semiprime, R^{(+)} is nondegenerate by (0.7)(i), and Q_{R^{−}}R^{+} ≠ 0. Moreover, Q_{R^{−}}R^{+} = L^{−}, where L is the ideal (R^{(+)})^{3} of R^{(+)}. Thus Q_{R^{−}}R^{+} is a nonzero semiideal of R^{(+)} and I = (Q_{R^{+}}Q_{R^{−}}R^{+}, Q_{R^{−}}R^{+}) is a nonzero ideal of R^{(+)} by [17, 6.2(b)] applied to T (R^{(+)}). By (1.5), there exists a nonzero ideal B of R contained in I, hence H ⊆ I since H ⊆ B.

4.7 Corollary. Let R be an associative pair such that R_{a}^{σ} is a simple algebra for any 0 ≠ a ∈ R^{−σ}, σ = ±. Then Heart R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}) is a simple associative pair.

4.8 Corollary. Let R be an associative pair. Then the following are equivalent:

(i) R is simple,

(ii) R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}) and R_{a}^{σ} is simple for any 0 ≠ a ∈ R^{−σ}, σ = ±,

(iii) R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}), R is semiprime and R_{a}^{+} is simple for any 0 ≠ a ∈ R^{−},

(iv) R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}), R is semiprime and R_{a}^{−} is simple for any 0 ≠ a ∈ R^{+}.

Proof: (i) ⟹ (ii) follows by [9, 5.2(iii)] and (4.7) since a simple pair must coincide with its heart.

(ii) ⟹ (iii) is obvious

(iii) ⟹ (i): By (4.6) Heart R = (Q_{R^{+}}Q_{R^{−}}R^{+}, Q_{R^{−}}R^{+}) is simple. But R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}) implies R^{−} = Q_{R^{−}}R^{+}, hence

R = (Q_{R^{+}}R^{−}, Q_{R^{−}}R^{+}) = (Q_{R^{+}}Q_{R^{−}}R^{+}, Q_{R^{−}}R^{+}) = Heart R.

The equivalence between (i), (ii) and (iv) follows as above, exchanging + and −.

4.9 Corollary. An associative algebra R is simple if and only if R = U_{R}R and R_{a} is simple for any 0 ≠ a ∈ R.

Proof: Use (4.8) with the fact that R is simple if and only if V (R) is simple [5, 3.1].

4.10 Remark: For an associative system R under the hypothesis of (4.5), (4.6) or (4.7), a description of Heart R in terms of associative powers of R is not known.

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