Fountain-Gould Left Orders for Associative Pairs

José A. Anquela ^{1}, Teresa Cortés ^{1}, anque@pinon.ccu.uniovi.es, cortes@pinon.ccu.uniovi.es Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo, Spain

Miguel Gómez Lozano ^{2}, Mercedes Siles Molina ^{2} magomez@agt.cie.uma.es, mercedes@agt.cie.uma.es Departamento de Álgebra, Geometría, y Topología, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain

Abstract: In this paper we introduce a notion of weak Fountain-Gould left order for associative pairs and give a Goldie-like theory of associative pairs which are weak Fountain-Gould left orders in semiprime pairs coinciding with their socles.

Introduction

In 1990, Fountain and Gould [11] gave a notion of order in a ring not necessarily unital, and obtained in [12] a Goldie-like characterization of two-sided orders in semiprime rings with descending chain condition on principal one-sided ideals, i.e., in semiprime rings coinciding with their socles. Later on, Ánh and Márki extended these results to one-sided orders [2, 3]. In [8], local algebras are used to introduce a notion of order for associative pairs and describe such two-sided orders in semiprime pairs coinciding with their socles.

In our work we will give a pair version of weak Fountain-Gould one-sided orders and study associative pairs which are weak Fountain-Gould one-sided orders in semiprime pairs coinciding with their socles. In this sense, we remove two-sidedness of the results of [8]. At the present moment some work on Goldie theory of Jordan systems is being done [9, 10], so that having a suitable Goldie theory for associative systems could be of some help.

We will use envelopes and local algebras to translate pair problems into algebra ones; thus, already known results on Fountain-Gould orders in rings [3, 11, 13] will be very important in our work. We will also study the connections of our notion of order and other related notions given for associative pairs [8, 14].

Our paper is divided into three sections, apart from Section 0, devoted to outline some preliminary results and definitions. Section 1 deals with the algebra envelopes of associative pairs. These will be the “atmospheres” where pairs imbed and calculations

2000 Mathematics Subject Classification: 16P60, 16S90 are performed; they must be close enough to the pairs we deal with, but, on the other hand, they must be general enough to allow working with a pair and its quotient at the same time. Using algebra envelopes, we introduce in Section 2 the notion of weak Fountain-Gould left order for associative pairs. We also show that the quotients obtained via this notion are particular cases of those one-sided quotient pairs studied in [14]. This is used, together with local inverse lifting properties, to show that in most of the cases, our notion does not depend on the algebra envelope where the computations are done, for example, when one restricts to semiprime pairs. In the last section, we study weak Fountain-Gould left orders in semiprime pairs coinciding with their socles and characterize them in terms of chain conditions on left annihilators and local Goldie dimension.

1 Partially supported by the Ministerio de Ciencia y Tecnología and Fondos FEDER, BFM2001-1938-C02-02.

2 Partially supported by the Ministerio de Ciencia y Tecnología and Fondos FEDER, BFM2001-1938-C02-01, and the Junta de Andalucía, FQM 264.

We finally remark that, though we always work with left orders and left quotients, the results in this paper have their right-side analogues, with obvious changes in the definitions, just reversing products in the proofs or applying the left-side results to the opposite pairs and algebras.

0. Preliminaries

0.1 Throughout this paper Φ will denote a unital commutative associative ring of scalars, and all algebraic systems will be defined on modules over Φ. Recall that an associative pair over Φ is a pair (A^{+}, A^{−}) of Φ-modules together with Φ-trilinear maps

\begin{matrix}A^{σ} × A^{−σ} × A^{σ} & → & A^{σ} \\ (x, y, z) & ↦ & xyz \end{matrix}

satisfying the following identities:

uv(xyz) = u(vxy)z = (uvx)yz  (1)

for all u, x, z ∈ A^{σ}, v, y ∈ A^{−σ} and σ = ±.

The first example of an associative pair is given by V (A) := (A, A), for any associative algebra A, where the triple product of x, y, and z is given by the usual product xyz in the algebra A.

0.2 Given an associative pair A = (A^{+}, A^{−}), and elements x, z ∈ A^{σ}, y ∈ A^{−σ}, σ = ±, recall that left, middle and right multiplications are defined by:

λ(x, y)z = μ(x, z)y = ρ(y, z)x = xyz.  (1)

From (0.1)(1), for any x, u ∈ A^{σ}, y, v ∈ A^{−σ},

λ(x, y)λ(u, v) = λ(xyu, v) = λ(x, yuv),  (2)

and similarly

ρ(u, v)ρ(x, y) = ρ(x, yuv) = ρ(xyu, v).  (3)

As a consequence of (2) and (3), it is clear that the linear span of all operators T : A^{σ} → A^{σ} of the form T = λ(x, y), for (x, y) ∈ A^{σ} × A^{−σ}, or T = Id_{A^{σ}} is a unital associative algebra; it will be denoted by Λ(A^{σ}, A^{−σ}). Clearly A^{σ} is a left Λ(A^{σ}, A^{−σ})-module. Similarly, we define P(A^{−σ}, A^{σ}) as the linear span of all the right multiplications and the identity on A^{σ}, so that A^{σ} becomes a left P(A^{−σ}, A^{σ})- module. We define the left ideals L ⊂ A^{σ} of A as the Λ(A^{σ}, A^{−σ})-submodules of A^{σ}, and the right ideals R ⊂ A^{σ} as the P(A^{−σ}, A^{σ})-submodules. A two-sided ideal B ⊂ A^{σ} is both a left and a right ideal. An ideal I = (I^{+}, I^{−}) of A is a pair of two-sided ideals of A such that A^{σ}I^{−σ}A^{σ} ⊆ I^{σ}, σ = ±. A Φ-submodule M of A^{σ} is called an inner ideal of A if xA^{−σ}x ⊆ M, for any x ∈ M.

0.3 An associative pair A is semiprime if and only if nonzero ideals of A have nonzero cube [I ideal of A and I^{σ}I^{−σ}I^{σ} = 0, σ = ±, implies I = 0]. We will say that A is prime if A is semiprime and two nonzero ideals of A have nonzero intersection. An element a in A^{σ} generates the ideal I = (I^{+}, I^{−}) of A given by I^{σ} := Φa + A^{σ}A^{−σ}a + aA^{−σ}A^{σ} + A^{σ}A^{−σ}aA^{−σ}A^{σ} and I^{−σ} := A^{−σ}aA^{−σ}. This provides elemental characterizations of semiprimeness and primeness (see [1, 1.18]): A is semiprime if and only if A is nondegenerate (aA^{σ}a = 0, a ∈ A^{−σ} ⇒ a = 0, σ = ±); A is prime if and only if A is elementally prime (aA^{σ}b = 0, a, b ∈ A^{−σ} ⇒ a = 0 or b = 0, σ = ±).

0.4 If A is a subalgebra of a unital associative algebra Q, we say that A is a weak classical left order in Q (or Q is a weak classical left quotient algebra of A) if for any q ∈ Q, there exist a, b ∈ A, such that a is invertible in Q and q = a^{−1}b. We say that A is a classical left order in Q (or Q is a classical left quotient algebra of A) when it is a weak classical left order and every regular element in A is invertible in Q.

0.5 Given an associative pair A = (A^{+}, A^{−}) and a ∈ A^{−σ}, the Φ-module aA^{σ}a becomes an associative algebra, denoted A_{a}^{σ} and called the local algebra of A at a by defining the product

axa · aya = axaya, for any x, y ∈ A^{σ}.

Let A^{σ(a)} be the a-homotope of A, i.e., the Φ-module A with product x ·_{a} y := xay for any x, y ∈ A^{σ}. The set Ker_{A} a = Ker a := {x ∈ A^{σ} | axa = 0} is an ideal of A^{σ(a)} and the quotient A^{σ(a)}/ Ker a is isomorphic to A_{a}^{σ} (see [5, 8]).

1. Algebra Envelopes of Associative Pairs

1.1 Associative pairs are really “abstract off-diagonal Peirce spaces” of associative algebras: let E be a unital associative algebra, and consider the Peirce decomposition E = E_{11} ⊕ E_{12} ⊕ E_{21} ⊕ E_{22} of E with respect to an idempotent e ∈ E, i.e.,

E_{11} = eEe, E_{12} = eE(1 − e), E_{21} = (1 − e)Ee and E_{22} = (1 − e)E(1 − e).

From the Peirce multiplication rules, (E_{12}, E_{21}) is a subpair of V (E). Conversely, every associative pair A = (A^{+}, A^{−}) can be obtained in this way (see [16, 2.3]): let C be the Φ-submodule of B = End_{Φ}(A^{+}) × End_{Φ}(A^{−})^{op} spanned by e_{1} = (Id_{A^{+}}, Id_{A^{−}}) and all (λ(x, y), ρ(x, y)), and similarly, let D be the submodule of B^{op} spanned by e_{2} = (Id_{A^{+}}, Id_{A^{−}}) and all (ρ(y, x), λ(y, x)) where (x, y) ∈ A^{+} × A^{−}. By (0.2), these Φ-linear spans are really subalgebras. Clearly, A^{+} is an (C, D)-bimodule if we set

cx = c^{+}(x), xd = d^{+}(x)

for x ∈ A^{+} and c = (c^{+}, c^{−}) ∈ C, d = (d^{+}, d^{−}) ∈ D. Similarly, A^{−} is a (D, C)- bimodule. Now we define bilinear maps on A^{±} × A^{∓} with values in C, respectively, D, by

xy = (λ(x, y), ρ(x, y)), yx = (ρ(y, x), λ(y, x)).

Then it is easy to check that (C, A^{+}, A^{−}, D) is a Morita context which gives rise to a unital associative algebra E (cf. [16, 2.3]). If we set e = e_{1}, then the pair A = (A^{+}, A^{−}) is isomorphic to the associative pair (E_{12}, E_{21}). Moreover E_{11} (respectively, E_{22}) is spanned by e and all products x_{12}y_{21} (respectively, 1 − e and all products y_{21}x_{12}) for x_{12} ∈ E_{12}, y_{21} ∈ E_{21}, and has the property that

x_{11}E_{12} = E_{21}x_{11} = 0 ⟹ x_{11} = 0,  x_{22}E_{21} = E_{12}x_{22} = 0 ⟹ x_{22} = 0. (1)

1.2 Let A be the subalgebra of E generated by E_{12} ∪ E_{21}, i.e., A = E_{12} ⊕ E_{12}E_{21} ⊕ E_{21}E_{12} ⊕ E_{21}. It is immediate that A is an ideal of E. We will call A the standard envelope of the associative pair A, and write τ = (τ^{+}, τ^{−}) for the natural inclusion τ^{σ} : A^{σ} ⟶ A of A into A.

1.3 Let A be an associative pair, A be an associative algebra, and φ = (φ^{+}, φ^{−}), where φ^{σ} : A^{σ} ⟶ A is an injective Φ-linear map, σ = ±. We will say that A is a subpair of (A, φ), if

(i) φ^{+}(A^{+}) ∩ φ^{−}(A^{−}) = 0,

(ii) φ^{+}(A^{+})φ^{+}(A^{+}) = φ^{−}(A^{−})φ^{−}(A^{−}) = 0, and

(iii) φ : A ⟶ V (A) is a pair homomorphism (hence monomorphism).

When A is a subpair of (A, φ), (A, φ) will be called an envelope of A if φ^{+}(A^{+}) ∪ φ^{−}(A^{−}) generates A as an algebra, i.e.,

(iv) A = φ^{+}(A^{+}) + φ^{+}(A^{+})φ^{−}(A^{−}) + φ^{−}(A^{−})φ^{+}(A^{+}) + φ^{−}(A^{−}).

An envelope (A, φ) of A will be called tight if every nonzero ideal of A hits A [I ∩ (φ^{+}(A^{+}) ∪ φ^{−}(A^{−})) ≠ 0 for every nonzero ideal I of A]. We will say that (A, φ) and (Ã, φ̃) are isomorphic envelopes of A if there exists an algebra isomorphism ψ : A ⟶ Ã such that ψ ∘ φ^{σ} = φ̃^{σ}, σ = ±.

Notice that, for an associative pair A, the standard envelope (A, τ) of A is an envelope of A in the above sense.

1.4 Let A be an associative algebra. An element a ∈ A is said to be a left (respectively, right) total zero divisor if aA = 0 (respectively, Aa = 0). Similarly, an element a ∈ A^{σ} of an associative pair is called a left (respectively, right) total zero divisor if aA^{−σ}A^{σ} = 0 (respectively, A^{σ}A^{−σ}a = 0). A pair or an algebra not having nonzero left (respectively, right) total zero divisors will be said left (respectively, right) faithful. It is immediate that a semiprime algebra or pair is both left and right faithful.

1.5 Proposition. Let A be a left and right faithful (for example, semiprime) associative pair, and (A, φ) be an envelope of A. Then

I = {x ∈ φ^{+}(A^{+})φ^{−}(A^{−}) + φ^{−}(A^{−})φ^{+}(A^{+}) | xφ^{σ}(A^{σ}) = 0 = φ^{σ}(A^{σ})x, σ = ±}
= {x ∈ φ^{+}(A^{+})φ^{−}(A^{−}) | xφ^{+}(A^{+}) = 0 = φ^{−}(A^{−})x}+
+ {x ∈ φ^{−}(A^{−})φ^{+}(A^{+}) | xφ^{−}(A^{−}) = 0 = φ^{+}(A^{+})x}

is the biggest ideal of A not hitting φ(A), and satisfies IA = AI = 0. Moreover, if ϕ^{σ} : A^{σ} ⟶ A/I is given by ϕ^{σ}(x^{σ}) = φ^{σ}(x^{σ}) + I, σ = ±, then (A/I, ϕ) is an envelope of A isomorphic to the standard envelope of A.

Proof: From (1.3), it is clear that I is an ideal of A and that both definitions of I agree. Indeed, Ax = xA = 0 for any x ∈ I. Moreover (I ∩φ^{+}(A^{+}), I ∩φ^{−}(A^{−})) is an ideal of φ(A) consisting of two-sided total zero divisors, hence it is zero by faithfulness of φ(A) ≅ A. This means that ϕ^{σ} is injective for σ = ±, and it is straightforward that (A/I, ϕ) satisfies (1.3)(ii)–(iv). To show that (A/I, ϕ) is an envelope of A, we just need to check (1.3)(i). Indeed, an element in ϕ^{+}(A^{+}) ∩ ϕ^{−}(A^{−}) has the form φ^{+}(a^{+}) + I = φ^{−}(a^{−}) + I for some a^{σ} ∈ A^{σ}, σ = ±. Thus, there must be y ∈ I such that φ^{+}(a^{+}) = φ^{−}(a^{−}) + y, hence

φ^{+}(a^{+})φ^{−}(A^{−})φ^{+}(A^{+}) = (φ^{−}(a^{−}) + y)φ^{−}(A^{−})φ^{+}(A^{+}) = 0,

because φ satisfies (1.3)(ii). Therefore φ^{+}(a^{+}) = 0, by left faithfulness of φ(A).

Let L be an ideal of A not hitting φ(A). We will show that L ⊆ I: if x ∈ L, write x = x^{+} + y + z + x^{−}, where x^{σ} ∈ φ^{σ}(A^{σ}), σ = ±, y ∈ φ^{+}(A^{+})φ^{−}(A^{−}), z ∈ φ^{−}(A^{−})φ^{+}(A^{+}). By (1.3)(ii), φ^{+}(A^{+})xφ^{+}(A^{+}) = φ^{+}(A^{+})x^{−}φ^{+}(A^{+}) ∈ φ^{+}(A^{+}) ∩ L = 0, hence x^{−} = 0 by left and right faithfulness of φ(A). Similarly x^{+} = 0. But now, again by (1.3)(ii), xφ^{+}(A^{+}) = yφ^{+}(A^{+}) ∈ φ^{+}(A^{+}) ∩ L = 0, and, similarly xφ^{−}(A^{−}) = φ^{+}(A^{+})x = φ^{−}(A^{−})x = 0, which shows x ∈ I.

Let (Ã, τ) be the standard envelope of A. We can define a linear map ψ : A ⟶ Ã given by

ψ(φ^{+}(x^{+}) + ∑_{i} φ^{+}(y_{i}^{+})φ^{−}(y_{i}^{−}) + ∑_{j} φ^{−}(z_{j}^{−})φ^{+}(z_{j}^{+}) + φ^{−}(u^{−}))

= τ^{+}(x^{+}) ⊕ ∑_{i} τ^{+}(y_{i}^{+})τ^{−}(y_{i}^{−}) ⊕ ∑_{j} τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}) ⊕ τ^{−}(u^{−}),

for any x^{+}, y_{i}^{+}, z_{j}^{+} ∈ A^{+}, y_{i}^{−}, z_{j}^{−}, u^{−} ∈ A^{−}. Indeed, if

a = φ^{+}(x^{+}) + ∑_{i} φ^{+}(y_{i}^{+})φ^{−}(y_{i}^{−}) + ∑_{j} φ^{−}(z_{j}^{−})φ^{+}(z_{j}^{+}) + φ^{−}(u^{−}) = 0,

then 0 = φ^{−}(A^{−})aφ^{−}(A^{−}) = φ^{−}(A^{−})φ^{+}(x^{+})φ^{−}(A^{−}) = φ^{−}(A^{−}x^{+}A^{−}) by (1.3), thus A^{−}x^{+}A^{−} = 0 by the injectivity of φ^{−}, which implies x^{+} = 0 by left and right faithfulness of A; similarly u^{−} = 0; hence

0 = φ^{+}(A^{+})a = φ^{+}(A^{+})(∑_{j} φ^{−}(z_{j}^{−})φ^{+}(z_{j}^{+})) = ∑_{j} φ^{+}(A^{+})φ^{−}(z_{j}^{−})φ^{+}(z_{j}^{+})

= φ^{+}(∑_{j} A^{+}z_{j}^{−}z_{j}^{+}),

which implies ∑_{j} A^{+}z_{j}^{−}z_{j}^{+} = 0, and thus

0 = τ^{+}(∑_{j} A^{+}z_{j}^{−}z_{j}^{+}) = ∑_{j} τ^{+}(A^{+})τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}) = τ^{+}(A^{+})(∑_{j} τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}));

similarly (∑_{j} τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}))τ^{+}(A^{+}) = 0, which implies ∑_{j} τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}) = 0 by (1.2) and (1.1)(1); in a similar manner ∑_{i} τ^{+}(y_{i}^{+})τ^{−}(y_{i}^{−}) = 0, and we get that ψ is well defined.

It is clear that ψ is a surjective algebra homomorphism satisfying ψ ∘ φ^{σ} = τ^{σ}, σ = ±. By the very definition of ψ, an element a as above lies in Ker ψ if and only if a = ∑_{i} φ^{+}(y_{i}^{+})φ^{−}(y_{i}^{−}) + ∑_{j} φ^{−}(z_{j}^{−})φ^{+}(z_{j}^{+}) with ∑_{i} τ^{+}(y_{i}^{+})τ^{−}(y_{i}^{−}) ⊕ ∑_{j} τ^{−}(z_{j}^{−})τ^{+}(z_{j}^{+}) = 0, which is shown to be equivalent to aφ^{σ}(A^{σ}) = φ^{σ}(A^{σ})a = 0, σ = ±, using (1.1)(1). Thus Ker ψ = I, and we can define ψ̃ : A/I ⟶ Ã by ψ̃(a+I) = ψ(a), which turns out to be an algebra isomorphism satisfying ψ̃∘ϕ^{σ} = τ^{σ}, σ = ±.

1.6 Corollary. Let A be a left and right faithful (for example, semiprime) associative pair, and (A, φ) be an envelope of A. Then the following are equivalent:

(i) (A, φ) is tight on A,

(ii) A is left and right faithful,

(iii) (A, φ) is isomorphic to the standard envelope of A.

Proof: Apply (1.5) together with the obvious fact that the set of left (respectively, right) total zero divisors of an algebra is an ideal.

1.7 Remark: To simplify notation, from now on, when dealing with a subpair A of (A, φ) we will assume that A^{σ} ⊆ A, the maps φ^{σ} will be simply the inclusion maps, and will write A instead of (A, φ). This will also be applied to the particular case of (A, φ) being an envelope of A.

2. Weak Fountain-Gould Left Orders in Associative Pairs

In this section we introduce the notion of weak Fountain-Gould left order for associative pairs, inspired by Fountain-Gould’s definition of left orders for rings given in [11].

2.1 Let a be an element of an algebra R. We say that b in R is the group inverse of a if aba = a, bab = b, and ab = ba. It is easy to see that the group inverse is unique when it exists, and that a has a group inverse in R (say b) if and only if there exists an idempotent e in R such that a is an invertible element of eRe (indeed, e is unique, given by e = ab). An element a is said to be locally invertible if it has a group inverse, which will be denoted by a^{#}. If a is locally invertible, then any power a^{n} of a is also locally invertible, and (a^{n})^{#} = (a^{#})^{n}. It is obvious that an invertible element a in a unital algebra is locally invertible with a^{#} = a^{−1}.

2.2 A subalgebra R of an associative algebra Q is said to be a weak Fountain-Gould left order in Q (or Q is said to be a weak Fountain-Gould left quotient algebra of R) if for every element q ∈ Q there exist a, x ∈ R such that a is locally invertible in Q and q = a^{#}x. Clearly, if R is a weak classical left order in a unital algebra Q, then R is a weak Fountain-Gould left order in Q. The converse is not true in general, as it was shown by Fountain and Gould in [11, Example 3.1].

A subalgebra R of an associative algebra Q is said to be a Fountain-Gould left order in Q (or Q is said to be a Fountain-Gould left quotient algebra of R) if A is a weak Fountain-Gould left order in Q and every square-cancellable element of R is locally invertible in Q (cf. [11]).

2.3 Let A be a subpair of an associative pair Q, which is a subpair of the algebra Q (in particular A is also a subpair of the algebra Q). We will say that A is a weak Fountain-Gould left order in Q relative to Q if for any q ∈ Q^{σ} there exist a ∈ A^{σ}A^{−σ} which is locally invertible in Q^{σ}Q^{−σ}, and b ∈ A^{σ}, such that q = a^{#}b. We will also say that Q is a weak Fountain-Gould left quotient pair of A relative to Q.

2.4 Remark: Under the conditions of (2.3), we can always replace Q by the subalgebra generated by Q^{+} ∪ Q^{−}, and assume that Q is an envelope of Q.

2.5 Remark: If A is a weak Fountain-Gould left order in Q relative to Q, for any q ∈ Q^{σ} we can always find a, b as in (2.3) such that also aq = b [just replace a and b in (2.3) by a^{2} and ab, respectively].

2.6 Examples: (i) Let R be a subalgebra of an associative algebra Q. It is easy to see that Q is a weak Fountain-Gould left quotient algebra of R if and only if V (Q) is a weak Fountain-Gould left quotient pair of the associative pair V (R) relative to the algebra of 2×2 matrices over Q with the natural imbedding

a ∈ V (Q)^{+} ↦ ( \begin{matrix}0 & a \\ 0 & 0 \end{matrix} ) ∈ M_{2×2}(Q), b ∈ V (Q)^{−} ↦ ( \begin{matrix}0 & 0 \\ b & 0 \end{matrix} ) ∈ M_{2×2}(Q).

(ii) Let D be a weak classical left order in a division algebra Δ. Then, one can show that, for every pair (m, n) of positive integers, (M_{m×n}(D), M_{n×m}(D)) is a weak Fountain-Gould left order in the associative pair (M_{m×n}(Δ), M_{n×m}(Δ)) relative to the algebra M_{(m+n)×(m+n)}(Δ), considering the natural imbedding given by

a ∈ M_{m×n}(Δ) ↦ ( \begin{matrix}0 & a \\ 0 & 0 \end{matrix} ) ∈ M_{(m+n)×(m+n)}(Δ),

b ∈ M_{n×m}(Δ) ↦ ( \begin{matrix}0 & 0 \\ b & 0 \end{matrix} ) ∈ M_{(m+n)×(m+n)}(Δ).

2.7 The following notion was introduced by Utumi in [18]: let R be a subalgebra of an associative algebra Q. We say that Q is a (general) left quotient algebra of R if for every x, y ∈ Q with x ≠ 0 there is a ∈ R such that ax ≠ 0 and ay ∈ R. Notice that an algebra is a left quotient algebra of itself if and only if it is right faithful. If R is right faithful, then it has a unique maximal left quotient algebra Q_{max}^{l}(R), which is unital, called the Utumi left quotient algebra of R [18].

2.8 In [14] a pair version of the above notion is introduced. Let A = (A^{+}, A^{−}) be a subpair of an associative pair Q = (Q^{+}, Q^{−}). We say that Q is a left quotient pair of A if given p, q ∈ Q^{σ} with p ≠ 0, σ = ±, there exist a ∈ A^{−σ}, b ∈ A^{σ} such that

bap ≠ 0 and baq ∈ A^{σ}.

2.9 Let A be an associative pair and let X ⊂ A^{σ}, σ = ±. The left annihilator of X in A is defined to be the set

lan(X) = lan_{A}(X) = {b ∈ A^{−σ} | bXA^{−σ} = 0, A^{σ}bX = 0}.

Similarly, the right annihilator of X in A is defined by

ran(X) = ran_{A}(X) = {b ∈ A^{−σ} | XbA^{σ} = 0, A^{−σ}Xb = 0}.

Clearly, lan(X) is a left ideal of A and ran(X) is a right ideal of A.

Let A be an associative pair which is a subpair of the algebra A. For σ = ± and an element a ∈ A^{σ}A^{−σ} we can also consider

lan_{A}(a) = {x ∈ A^{−σ} | xa = 0}, (respectively, ran_{A}(a) = {x ∈ A^{σ} | ax = 0})

which is a left (respectively, right) ideal of A.

2.10 Lemma. Let A be a subpair of an associative pair Q, which is a subpair of the algebra Q, and let a ∈ A^{σ}A^{−σ} satisfy that there exists a^{#} ∈ Q^{σ}Q^{−σ}. Then:

(i) Q^{−σ} = Q^{−σ}a + lan_{Q}(a).

If A is a weak Fountain-Gould left order in Q relative to Q, then:

(ii) lan_{Q}(a) = Q^{−σ}Q^{σ} lan_{A}(a).
(iii) Q^{−σ} = Q^{−σ}Q^{σ}(lan_{A}(a) + A^{−σ}a).

Proof: (i) If q ∈ Q^{−σ}, then q = qa^{#}a + (q − qa^{#}a), where qa^{#}a ∈ Q^{−σ}a and (q − qa^{#}a) ∈ lan_{Q}(a), and we have shown Q^{−σ} ⊆ Q^{−σ}a + lan_{Q}(a). The converse is obvious.

(ii) Let q ∈ lan_{Q}(a) and write q = u^{#}v, with u ∈ A^{−σ}A^{σ} (u^{#} ∈ Q^{−σ}Q^{σ}), v ∈ A^{−σ}, and uq = v. Then v ∈ lan_{Q}(a)∩A^{−σ} = lan_{A}(a) and q = u^{#}v ∈ Q^{−σ}Q^{σ} lan_{A}(a). The converse is obvious.

(iii) follows from (i), (ii), and the fact that Q^{−σ}Q^{σ}A^{−σ} = Q^{−σ}.

2.11 Proposition. If A is a weak Fountain-Gould left order in an associative pair Q relative to an algebra Q, then Q is a left quotient pair of A.

Proof: Let p, q ∈ Q^{σ} with p ≠ 0. We can write q = a^{#}b, p = c^{#}d where a, c ∈ A^{σ}A^{−σ} (a^{#}, c^{#} ∈ Q^{σ}Q^{−σ}) and b, d ∈ A^{σ}. By (2.5), we can also assume that aq = b. Notice that 0 ≠ p = c^{#}d = c^{#}cc^{#}d = c^{#}cp ∈ Q^{σ}Q^{−σ}p. Hence 0 ≠ Q^{σ}Q^{−σ}p = Q^{σ}Q^{−σ}Q^{σ}(lan_{A}(a) + A^{−σ}a)p by (2.10)(iii), and there exist t ∈ Q^{σ}, x ∈ lan_{A}(a) and y ∈ A^{−σ} such that 0 ≠ t(x + ya)p. If we write t = u^{#}v, with u ∈ A^{σ}A^{−σ} (u^{#} ∈ Q^{σ}Q^{−σ}) and v ∈ A^{σ}, then v(x + ya)p ≠ 0 and

v(x + ya)q = vxq + vyaq = vxa^{#}b + vyaq = vxaa^{#}a^{#}b + vyaq
= vyaq (xa = 0 since x ∈ lan_{A}(a)) = vyb ∈ A^{σ}.

Next we will show that, for pairs without total zero divisors, the notion of weak Fountain-Gould left order is intrinsic, independent of the “atmosphere” algebra Q we deal with.

2.12 Lemma. Let Q be a left quotient pair of an associative pair A. Then A and Q are both right faithful. Moreover:

(i) If A is left faithful, then Q is left faithful.

(ii) If A is semiprime, then Q is semiprime.

Proof: Let 0 ≠ q ∈ Q^{σ}. By the very definition of left quotient pair, we can find a ∈ A^{σ}, b ∈ A^{−σ} such that 0 ≠ abq ∈ A^{σ}. But abq ∈ A^{σ}A^{−σ}q ⊆ Q^{σ}Q^{−σ}q, which shows the first assertion.

(i) If qQ^{−σ}Q^{σ} = 0, then abqA^{σ}A^{−σ} ⊆ abqQ^{−σ}Q^{σ} = 0, which is a contradiction if A is left faithful.

(ii) is [14, 2.13(ii)].

2.13 Lemma. Let R be an associative algebra, and I be an ideal of R such that RI = IR = 0. If x ∈ R/I is locally invertible in R/I with group inverse x^{#} ∈ R/I, then there exists a locally invertible element a ∈ R such that x = a + I and x^{#} = a^{#} + I. Indeed, if x = u + I, and x^{#} = v + I, we can set a = uvu, a^{#} = vuv.

Proof: Let x = u + I, x^{#} = v + I, where u, v ∈ R. By the definition of group inverse, we obtain

uv − vu ∈ I,  uvu − u ∈ I,  vuv − v ∈ I  (1)

Let a = uvu, a′ = vuv. By (1),

x = u + I = uvu + I = a + I, x^{#} = v + I = vuv + I = a′ + I.

Moreover,

a′a = vuvuvu = vu(vuv − v)u + vuvu = vuvu [since vu(vuv − v)u ∈ RIR = 0]

= vu(vu − uv) + vuuv = vuuv [since vu(vu − uv) ∈ RI = 0]
= (vu − uv)uv + uvuv = uvuv [since (vu − uv)uv ∈ IR = 0]

= (u − uvu)vuv + uvuvuv = uvuvuv [since (u − uvu)vuv ∈ IR = 0] = aa′.

Similarly, a′aa′ = a and aa′a = a, which shows a′ is the group inverse of a.

2.14 Proposition. Let A be a subpair of an associative pair Q such that either A or Q is left faithful (for example, if either A or Q is semiprime), and let Q be an associative algebra such that Q is a subpair of Q. Then A is a weak Fountain-Gould left order in Q relative to Q if and only if A is a weak Fountain-Gould left order in Q relative to the standard envelope Q̃ of Q.

Proof: Notice that (2.11) and (2.12) imply that Q is left and right faithful in any case. By (2.4), we can assume that Q is an envelope of Q, and, by (1.5) there exists an ideal I of Q not hitting Q such that Q/I and Q̃ are isomorphic envelopes of Q, and QI = IQ = 0.

If A is a weak Fountain-Gould left order in Q relative to Q, then it is straightforward that A is a weak Fountain-Gould left order in Q relative to Q/I, i.e., A is a weak Fountain-Gould left order in Q relative to Q̃.

Conversely, assume that A is a weak Fountain-Gould left order in Q relative to Q̃, i.e., A is a weak Fountain-Gould left order in Q relative to Q/I. Thus, given q ∈ Q^{σ}, there exist u ∈ A^{σ}A^{−σ}, b ∈ A^{σ} such that u + I is locally invertible, (u + I)^{#} = v + I with v ∈ Q^{σ}Q^{−σ}, and q + I = (u + I)^{#}(b + I). By (2.13), a = uvu ∈ Q^{σ}Q^{−σ} is locally invertible with a^{#} = vuv ∈ Q^{σ}Q^{−σ}, and u + I = a + I, (u + I)^{#} = v + I = a^{#} + I. Thus q − a^{#}b ∈ I ∩ Q^{σ} = 0, i.e., q = a^{#}b. The problem is that we do not know whether a is in A^{σ}A^{−σ}. Anyway,

a^{2} = au + a(a − u) = au [since a(a − u) ∈ QI = 0]
= u^{2} + (a − u)u = u^{2} [since (a − u)u ∈ IQ = 0] ∈ A^{σ}A^{−σ},
ab = ub + (a − u)b = ub [since (a − u)b ∈ IQ = 0] ∈ A^{σ}A^{−σ}A^{σ} ⊆ A^{σ},

and q = a^{#}b = (a^{#})^{2}ab, where (a^{2})^{#} = (a^{#})^{2} ∈ Q^{σ}Q^{−σ}.

2.15 Remark: Under the conditions of (2.14), there is no need to specify the algebra Q where the calculations are made, and we will simply say that A is a weak Fountain-Gould left order in Q.

3. Weak Fountain-Gould Left Orders in Semiprime Pairs Coinciding with their Socles

3.1 For an associative pair, different reasonable notions of socle can be considered (see [6]). However, we will restrict to semiprime pairs, where these notions coincide [6, 3.7]: given a semiprime associative pair A the socle Soc(A) of A is the pair (Soc(A)^{+}, Soc(A)^{−}), where Soc(A)^{σ} is the sum of all minimal left ideals of A contained in A^{σ} (Soc(A)^{σ} = 0 if A does not have minimal left ideals contained in A^{σ}), and Soc(A)^{σ} coincides with the sum of all minimal right or inner ideals of A contained in A^{σ}.

3.2 Lemma. Let Q be an associative pair, and Q be its standard envelope. Then Q is semiprime if and only if Q is semiprime.

Proof: Let E be the associative algebra built out of Q, as in (1.1), so that Q is an ideal of E (1.2).

If Q is semiprime, then E is semiprime by [8, 4.2] and Q is semiprime because it is and ideal of E. Conversely, if Q is semiprime, we will show that Q is so. Indeed, let us show that Q is nondegenerate (0.3): if q ∈ Q^{σ} satisfies 0 = qQ^{−σ}q, then qQq = q(Q^{σ} + Q^{−σ} + Q^{σ}Q^{−σ} + Q^{−σ}Q^{σ})q = qQ^{−σ}q = 0 since qQ^{σ}, Q^{σ}q ⊆ Q^{σ}Q^{σ} = 0; thus q = 0 by semiprimeness (nondegeneracy) of Q.

3.3 Lemma. Let Q be a semiprime associative pair, Q its standard envelope. Then Soc(Q)^{σ} = Soc(Q) ∩ Q^{σ}. Moreover, Q = Soc(Q) if and only if Q = Soc(Q), and, in this case, Q^{σ}Q^{−σ} (σ = ±) is a semiprime algebra which coincides with its socle.

Proof: Let E be the associative algebra built out of Q, as in (1.1), so that Q is an ideal of E (1.2). By [8, 4.2], E is semiprime, hence Soc(Q) = Soc(E) ∩ Q using (3.1), and [15, Prop. 3] applied to V (E)^{(+)}, together with [6, 2.1, 4.2]. Thus Soc(Q) ∩ Q^{σ} = Soc(E) ∩ Q ∩ Q^{σ} = Soc(E) ∩ Q^{σ} = Soc(Q)^{σ} by [8, 4.7(i)].

Obviously, the above equality shows Q = Soc(Q) when Q = Soc(Q). If, conversely, Q = Soc(Q), then Soc(Q) is an ideal of Q, containing Q^{+} ∪ Q^{−}, hence Q = Soc(Q) since Q^{+} ∪ Q^{−} generate Q as an algebra.

Assume now that Q = Soc(Q), and let e be the idempotent of E given in (1.1). Now eEe, (1 − e)E(1 − e) are semiprime algebras (using (0.3)), and Q^{+}Q^{−} = Q ∩ eEe, Q^{−}Q^{+} = Q ∩ (1 − e)E(1 − e) are ideals of them, hence semiprime. Moreover, again by (3.1) and [15, Prop. 3] we get

Soc(Q^{+}Q^{−}) = Soc(eEe) ∩ Q^{+}Q^{−}, Soc(Q^{−}Q^{+}) = Soc((1 − e)E(1 − e)) ∩ Q^{−}Q^{+}. (1)

On the other hand, [15, Prop. 3] also applies to full subpairs, which, again together with (3.1), yields

Soc(eEe) = Soc(E) ∩ eEe, Soc((1 − e)E(1 − e)) = Soc(E) ∩ (1 − e)E(1 − e). (2)

Finally, using (1) and (2),

Soc(Q^{+}Q^{−}) = Soc(eEe) ∩ Q^{+}Q^{−} = Soc(E) ∩ eEe ∩ Q^{+}Q^{−}
= Soc(E) ∩ Q^{+}Q^{−} ⊇ Soc(Q) ∩ Q^{+}Q^{−} = Q ∩ Q^{+}Q^{−} = Q^{+}Q^{−},

hence Soc(Q^{+}Q^{−}) = Q^{+}Q^{−}, and, similarly, Soc(Q^{−}Q^{+}) = Q^{−}Q^{+}.

3.4 Theorem. Let A be a weak Fountain-Gould left order in a simple associative pair Q coinciding with its socle. Then A is a prime associative pair.

Proof: By (2.14), we may assume that A is a weak Fountain-Gould left order in Q relative to the standard envelope Q of Q.

Let a, c be nonzero elements of A^{σ}. Since Q is simple, it is prime. Hence there is an element q ∈ Q^{−σ} satisfying aqc ≠ 0 (0.3). On the other hand, q = u^{#}v, where u ∈ A^{−σ}A^{σ} (u^{#} ∈ Q^{−σ}Q^{σ}), v ∈ A^{−σ}, and uu^{#}v = v (2.5). Then 0 ≠ aqc = au^{#}vc. By semiprimeness of Q (0.3), au^{#}vcQ^{−σ} ≠ 0 and, since Q = Soc(Q), there exists an element t in a minimal right ideal I of Q contained in Q^{−σ}, satisfying 0 ≠ au^{#}vct ∈ Q^{−σ}Q^{σ}.

Let us consider the set Γ = {u^{n}u^{#}vctQ^{σ} | n = 1, 2, …} of right ideals of V := V (Q^{−σ}Q^{σ}) contained in uV^{−σ}V^{σ} = uQ^{−σ}Q^{σ}Q^{−σ}Q^{σ}. Notice that all the right ideals in Γ are nonzero (0 ≠ u^{#}vct = (u^{#})^{n}u^{n}u^{#}vct ⟹ u^{n}u^{#}vct ≠ 0 ⟹ u^{n}u^{#}vctQ^{σ} ≠ 0 by semiprimeness of Q). On the other hand, Q^{−σ}Q^{σ} is a semiprime algebra coinciding with its socle by (3.3), so that V is a semiprime pair coinciding with its socle [6, 2.1, 4.2]. Hence, by [8, 5.2(vi)], the sum of the elements of Γ cannot be direct, and so there exist α_{1}, …, α_{m} ∈ Q^{σ} such that

∑_{i=1}^{m} u^{i}u^{#}vctα_{i} = 0  (1)

and at least one summand is nonzero. Let u^{k}u^{#}vctα_{k} be the first nonzero summand. By multiplying (1) by (u^{#})^{k} on the left and renumbering the α’s, we obtain

u^{#}vctα_{1} + uu^{#}vctα_{2} + … + u^{r−1}u^{#}vctα_{r} = 0,  (2)

with u^{#}vctα_{1} ≠ 0. On the other hand, tα_{1} is nonzero element in the semiprime algebra Q^{−σ}Q^{σ}, hence tα_{1}Q^{−σ}Q^{σ} ≠ 0, which implies that tα_{1}Q^{−σ} is a nonzero right ideal of Q contained in I. By minimality, I = tα_{1}Q^{−σ}, hence t = tα_{1}α for some α ∈ Q^{σ}. Multiplying (2) by a on the left and by α on the right we obtain:

au^{#}vctα_{1}α + auu^{#}vctα_{2}α + … + au^{r−1}u^{#}vctα_{r}α = 0,

i.e.,

au^{#}vct + auu^{#}vctα_{2}α + … + au^{r−1}u^{#}vctα_{r}α = 0.  (3)

Then

\begin{aligned}0 ≠ −au^{#}vct & = auu^{#}vctα_{2}α + … + au^{r−1}u^{#}vctα_{r}α \\ & = avctα_{2}α + … + au^{r−2}vctα_{r}α \end{aligned}  (4)

since uu^{#}v = v. Thus au^{k}vctα_{k+2}α ≠ 0 for some k ∈ {0, …, r−2} and, consequently, 0 ≠ au^{k}vc ∈ aA^{−σ}c, which proves that A is prime (0.3).

3.5 Corollary. Let A be a weak Fountain-Gould left order in a semiprime associative pair Q coinciding with its socle. Then A is semiprime.

Proof: By [7, Theorem 1], Q is a direct sum of ideals Q_{i} (i ∈ I), where Q_{i} is simple and coincides with its socle. For every i ∈ I, let π_{i} = (π_{i}^{+}, π_{i}^{−}) : Q → Q_{i} be the canonical projection. It is easy to see that π_{i}(A) is a weak Fountain-Gould left order in Q_{i} and, by (3.4), we have that π_{i}(A) is a prime algebra, for any i ∈ I.

We will show that A is nondegenerate (0.3): given 0 ≠ x ∈ A^{σ}, there exists j ∈ I such that π_{j}^{σ}(x) ≠ 0. By primeness of π_{j}(A), 0 ≠ π_{j}^{σ}(x)(π_{j}(A))^{−σ}π_{j}^{σ}(x) = π_{j}^{σ}(x)π_{j}^{−σ}(A^{−σ})π_{j}^{σ}(x) = π_{j}^{σ}(xA^{−σ}x), which implies xA^{−σ}x ≠ 0.

In the next result we express the fact of A being a weak Fountain-Gould left order in a semiprime pair Q coinciding with its socle in terms of the algebras naturally attached to these pairs.

3.6 Remark. We claim that [8, 8.7, 8.8] remain true when replacing “A being a left triple product order in Q” by “Q being a left quotient pair of A”. Indeed, the proofs given in [8] remain valid under this new condition, which can be readily checked (also cf. [14, 2.14]).

3.7 Theorem. Let A be a subpair of a semiprime associative pair Q which coincides with its socle, and Q be the standard envelope of Q. Let A be the subalgebra of Q generated by A^{+} ∪ A^{−}, i.e., A = A^{+} + A^{+}A^{−} + A^{−}A^{+} + A^{−}. The following are equivalent conditions:

(i) A is a weak Fountain-Gould left order in Q.

(ii) A is semiprime, Q^{σ} = A^{σ}Q^{−σ}A^{σ}, σ = ±, and Q is a left quotient pair of A.

(iii) A is semiprime, Q^{σ} = A^{σ}Q^{−σ}A^{σ} and, for every a ∈ A^{−σ}, A_{a}^{σ} is a classical left order in Q_{a}^{σ}, σ = ±.

(iv) A is a Fountain-Gould left order in Q.

(v) A^{σ}A^{−σ} is a Fountain-Gould left order in Q^{σ}Q^{−σ}, for σ = ±.

Moreover, under any of the above conditions, Q^{σ} = Q^{σ}A^{−σ}A^{σ} = A^{σ}A^{−σ}Q^{σ}, σ = ±, and A is isomorphic to the standard envelope of A.

Proof: (i)⇒(ii) By (3.5), A is a semiprime associative pair. For any q ∈ Q^{σ}, there exist a ∈ A^{σ}A^{−σ}, b ∈ A^{σ}, with a^{#} ∈ Q^{σ}Q^{−σ} such that q = a^{#}b = a(a^{#})^{2}b ∈ A^{σ}A^{−σ}Q^{σ}Q^{−σ}A^{σ} = A^{σ}(A^{−σ}Q^{σ}Q^{−σ})A^{σ} ⊆ A^{σ}Q^{−σ}A^{σ}, and Q is a left quotient pair of A by (2.11). Moreover, q = a^{#}b = (a^{#})^{2}ab ∈ Q^{σ}Q^{−σ}A^{σ}A^{−σ}A^{σ} = (Q^{σ}Q^{−σ}A^{σ})A^{−σ}A^{σ} ⊆ Q^{σ}A^{−σ}A^{σ}, and q = a^{#}b = a(a^{#})^{2}b ∈ A^{σ}A^{−σ}Q^{σ}Q^{−σ}A^{σ} = A^{σ}A^{−σ}(Q^{σ}Q^{−σ}A^{σ}) ⊆ A^{σ}A^{−σ}Q^{σ}.

(ii)⇒(iii) follows from [8, 8.8(ii)] and (3.6).

(iii)⇒(ii) First we claim that:

(1) If 0 ≠ aqc, where a, c ∈ A^{−σ}, q ∈ Q^{σ}, then the set Λ_{aqc} = {aba ∈

A_{a}^{σ} | abaqc ∈ A^{−σ}} is an essential left ideal of A_{a}^{σ}.

Indeed Λ_{aqc} is clearly a left ideal of A_{a}^{σ}. Any nonzero left ideal of A_{a}^{σ} has the form aIa, where I ⊆ A^{σ}. If aIaqc = 0, then aIa ⊆ Λ_{aqc}, hence aIa hits Λ_{aqc}. Assume y ∈ I satisfies ayaqc ≠ 0. By semiprimeness of Q (0.3), ayaqcpayaqc ≠ 0 for some p ∈ Q^{σ}, hence cpayaqc ≠ 0. Since A_{c}^{σ} is a classical left order in Q_{c}^{σ}, cpayaqc = cūcvc, for some elements cuc, cvc ∈ A_{c}^{σ}, such that cuc is regular in A_{c}^{σ}, and cūc is the inverse of cuc in Q_{c}^{σ}. Then

0 ≠ cucpayaqc = cvc ∈ A_{c}^{σ}.

By [7, Theorem 1], Q is a direct sum of ideals Q_{α} (α ∈ J), where the Q_{α}’s are simple and coincide with their socles. Let π_{α} = (π_{α}^{+}, π_{α}^{−}) : Q → Q_{α} be the canonical projection, α ∈ J. For every w ∈ A^{−σ}, π_{α}^{−σ} induces μ_{α} : Q_{w}^{σ} ⟶ Q_{π_{α}^{−σ}(w)}^{σ} = (Q_{α})_{π_{α}^{−σ}(w)}^{σ}, an algebra epimorphism . Now, [13, 4.6] applies to obtain that μ_{α}(A_{w}^{σ}) = π_{α}^{σ}(A^{σ})_{π_{α}^{−σ}(w)} is a classical left order in (Q_{α})_{π_{α}^{−σ}(w)}^{σ}, whenever π_{α}^{−σ}(w) ≠ 0. In particular π_{α}^{σ}(A^{σ})_{π_{α}^{−σ}(w)} ≠ 0 if π_{α}^{−σ}(w) ≠ 0, which implies that π_{α}(A) is semiprime (0.3). Since Q_{α} is simple for any α ∈ J, by [8, 5.2(iii)(v)], (Q_{α})_{π_{α}^{−σ}(w)}^{σ} is a simple artinian algebra, and, by Goldie’s First Theorem [17, 3.2.16], π_{α}^{σ}(A^{σ})_{π_{α}^{−σ}(w)} is a prime algebra, whenever 0 ≠ π_{α}^{−σ}(w), which implies that π_{α}(A) is prime, using [8, 5.2(ii)]. Choose β ∈ J, such that 0 ≠ π_{β}^{−σ}(cvc) = π_{β}^{−σ}(cucpayaqc). Then π_{β}^{−σ}(a) ≠ 0, and, by primeness (0.3) of π_{β}(A), there exists x ∈ A^{σ} such that 0 ≠ π_{β}^{−σ}(a)π_{β}^{σ}(x)π_{β}^{−σ}(cvc) = π_{β}^{−σ}(axcvc). In particular,

axcvc = axcucpayaqc ≠ 0.  (2)

Take ara, asa ∈ A_{a}^{σ}, such that ara is regular in A_{a}^{σ} and axcucpa = ar̄asa, where ar̄a denotes the inverse of ara in Q_{a}^{σ}. Then araxcucpa = asa, and (2) yields

0 ≠ araxcucpayaqc = araxcvc ∈ araxcA^{σ}c ⊆ A^{−σ}

and araxcucpaya = asaya is a nonzero element in Λ_{aqc} ∩ aIa, which proves (1).

Now we can prove that Q is a left quotient pair of A: let p, q be in Q^{σ}, with p ≠ 0. Since Q is semiprime Q^{σ}Q^{−σ}p ≠ 0 and since Q^{σ} = A^{σ}Q^{−σ}A^{σ}, σ = ±, then 0 ≠ Q^{σ}Q^{−σ}p = Q^{σ}A^{−σ}Q^{σ}A^{−σ}p = Q^{σ}A^{−σ}A^{σ}Q^{−σ}A^{σ}A^{−σ}p. Hence there exist a ∈ A^{σ}, b ∈ A^{−σ} such that abp ≠ 0. If abq = 0, we are done. Otherwise, we can use the fact that q ∈ Q^{σ} = A^{σ}Q^{−σ}A^{σ} to find q_{j} ∈ Q^{−σ}, d_{j} ∈ A^{σ} such that abq = ∑_{j=1}^{n} aq_{j}d_{j}, and aq_{j}d_{j} ≠ 0 for any j = 1, …, n. By (1), the Λ_{aq_{j}d_{j}}’s are essential left ideals of A_{a}^{−σ} and, consequently, Λ = (∩_{j=1}^{n}Λ_{aq_{j}d_{j}}) is an essential left ideal of A_{a}^{−σ}. Applying Goldie’s Second Theorem [17, 3.2.14], there exists aua ∈ Λ such that aua is regular in A_{a}^{−σ}. Then auabq = ∑_{j=1}^{n} auaq_{j}d_{j} ∈ A^{σ} because aua ∈ Λ_{aq_{j}d_{j}}for every j, and auabp ≠ 0 (0 ≠ abp ⟹ 0 ≠ abpQ^{−σ}abp by semiprimeness, but abpQ^{−σ}abp = aūauabpQ^{−σ}abp, because aūa is the inverse of aua in Q_{a}^{−σ}).

(ii)⇒(iv) By (3.2), the algebra Q is semiprime, and by (3.3), it coincides with its socle. Since A is semiprime, by the proof of [14, 2.5(i)(iii)] Q is a left quotient algebra of A, and this latter is isomorphic to the standard envelope of A, which is semiprime using (3.2). The equality Q^{σ} = A^{σ}Q^{−σ}A^{σ} readily implies Q = AQ, hence A is a Fountain-Gould left order in Q by [13, 4.11].

(iv)⇒(v) By (3.3), Q^{σ}Q^{−σ} is semiprime and coincides with its socle. Moreover, by A being a Fountain-Gould left order in Q, we have

Q = AQ,  (1)

which yields

Q^{σ} = A^{σ}Q^{−σ}Q^{σ} + A^{σ}A^{−σ}Q^{σ} ⊆ A^{σ}Q^{−σ}Q^{σ}.  (2)

Hence

Q^{σ}Q^{−σ} = A^{σ}Q^{−σ} + A^{σ}A^{−σ}Q^{σ}Q^{−σ} (by (1)) ⊆ A^{σ}A^{−σ}Q^{σ}Q^{−σ} ⊆ Q^{σ}Q^{−σ} (3)

by (2). Similarly, Q = QA implies Q^{σ}Q^{−σ} = Q^{σ}Q^{−σ}A^{σ}A^{−σ}, which implies, using (3), that

Q^{σ}Q^{−σ} = A^{σ}A^{−σ}Q^{σ}Q^{−σ}A^{σ}A^{−σ}.

For every 0 ≠ a ∈ A^{σ}A^{−σ}, (A^{σ}A^{−σ})_{a} = A_{a} is a classical left order in Q_{a} = (Q^{σ}Q^{−σ})_{a} [13, 4.7], hence A^{σ}A^{−σ} is a Fountain-Gould left order in Q^{σ}Q^{−σ} by [13, 4.11].

(v)⇒(i) Let p ∈ Q^{σ}. By (3.3), Q = Soc(Q). In particular Q is von Neumann regular [15, Th. 1], hence there exists q ∈ Q such that p = pqp. Notice that this equality also holds if we replace q by its component in Q^{−σ}, thus we can assume that q ∈ Q^{−σ}. Now, qp = a^{#}b for some a, b ∈ A^{−σ}A^{σ} such that a^{#} ∈ Q^{−σ}Q^{σ}. Let b_{i} ∈ A^{−σ} and c_{i} ∈ A^{σ} satisfy b = ∑_{i=1}^{n} b_{i}c_{i}. Then p = pqp = pa^{#}b = ∑_{i=1}^{n} pa^{#}b_{i}c_{i}. Using [3, Theorem 5], pa^{#}b_{i} = u^{#}v_{i}, i = 1, …, n, for some u, v_{i} ∈ A^{σ}A^{−σ} such that u^{#} ∈ Q^{σ}Q^{−σ}. Then p = u^{#}(∑_{i=1}^{n} v_{i}c_{i}), which proves (i).

Now we establish an analogue of [3, Theorem 5] for pairs. The way we state the result is due to the fact that, unlike the algebra case, having a common denominator for two elements does not seem to imply the property for an arbitrary finite family.

3.8 Corollary (Common Denominator Property). Let A be a weak Fountain-Gould left order in a semiprime associative pair Q which coincides with its socle. Then, given any finite number of elements p_{1}, …, p_{n} ∈ Q^{σ}, there exist a ∈ A^{σ}A^{−σ}, b_{1}, …, b_{n} ∈ A^{σ} such that a is locally invertible in Q^{σ}Q^{−σ}, and p_{i} = a^{#}b_{i}, for i = 1, …, n.

Proof: We can write p_{i} = (u_{i})^{#}v_{i}, with u_{i} ∈ A^{σ}A^{−σ} and v_{i} ∈ A^{σ}. By (3.7), A^{σ}A^{−σ} is a Fountain-Gould left order in Q^{σ}Q^{−σ}, and by [3, Theorem 5], given (u_{1})^{#}, …, (u_{n})^{#} there exist a, c_{1}, …, c_{n} ∈ A^{σ}A^{−σ} such that (u_{i})^{#} = a^{#}c_{i}, i = 1, …, n. Hence, if we define b_{i} := c_{i}v_{i}, we have p_{i} = a^{#}b_{i}, for i = 1, …, n.

We finally express the fact of A being a weak Fountain-Gould left order in a semiprime pair Q coinciding with its socle in intrinsic pair terms and Goldie theory notions [8].

3.9 Theorem. For an associative pair A the following conditions are equivalent:

(i) A is a weak Fountain-Gould left order in a semiprime associative pair Q coinciding with its socle.

(ii) A is semiprime, satisfies the ascending chain condition on lan_{A}(x), with x ∈ A^{σ}, σ = ±, and has finite left local Goldie dimension.

(iii) A is a semiprime left local Goldie associative pair.

(iv) A is semiprime and all its local algebras at nonzero elements are left Goldie.

(v) A is semiprime and there is a semiprime associative pair coinciding with its socle which is a left quotient pair of A.

In this case,

(1) A is prime if and only if Q is simple, and

(2) A is left Goldie if and only if Q is artinian.

Proof: (i)⇒(ii) By (3.5) A is semiprime. Moreover since Q is a left quotient pair by (2.11), it follows from [8, 8.8(i)] and (3.6) that A is left local Goldie, hence it has finite left local Goldie dimension. By [8, 8.7(ii), 2.5(1)(iii)] and (3.6), A satisfies the ascending chain condition on the left annihilators of a single element.

(ii)⇒(iii) is a consequence of [8, 3.6];

(iii)⇒(i) Let A be the standard envelope of A.

We will first show that A is a semiprime left local Goldie associative algebra: by (3.2) and [14, 1.9], the standard envelope A of A is semiprime and left nonsingular. On the other hand, for every a ∈ A^{σ}, the Goldie (or uniform) dimension u-dim_{A}(a) of a in A equals the uniform dimension u-dim(A_{a}) of A_{a} by [13, 2.1(iv)], and we remark A_{a}^{−σ} = A_{a}. Moreover, by [8, 5.2(iv)], u-dim(A_{a}^{−σ}) = u-dim_{A}(a) < ∞ since A is left local Goldie. Thus we have shown that u-dim_{A}(a) < ∞, for any a ∈ A^{σ}, i.e., A^{+} ∪ A^{−} ⊆ I(A), where I(A) denotes the set of elements of A of finite uniform dimension. But I(A) is an ideal of A by [4, Prop. 1; 14, 3.2(iii)], and A^{+} ∪ A^{−} generate A, hence the whole algebra A is contained in I(A), i.e., A = I(A).

Let Q = Q_{max}^{l}(A). By [13, 4.9(i)] A is a Fountain-Gould left order in AQ, which is a semiprime algebra coinciding with its socle.

Let E be the algebra built out of A, as in (1.1), having a unit element 1 and an idempotent e such that (A^{+}, A^{−}) = (eE(1 − e), (1 − e)Ee). By [14, 2.9(i)], Q = Q_{max}^{l}(E), and e, 1 − e ∈ Q. Let Q := (eAQ(1 − e), (1 − e)AQe). Notice that A ⊆ Q, and Q^{+} ∪ Q^{−} ⊆ AQ since eA, (1 − e)A ⊆ A.

We claim that:

(a) Q is semiprime and Q = Soc(Q).

Let x be a nonzero element in Q^{σ}, σ ∈ {+, −}. Since AQ is semiprime and coincides with its socle, (AQ)_{x} is a (nonzero) semiprime artinian algebra by [13, 2.1(i)(v)]. But Q_{x}^{−σ} = (AQ)_{x}, hence (a) follows by [8, 5.2(v)].

(b) A is a weak Fountain-Gould left order in Q.

By [14, 2.9, 2.11], Q_{max}^{l}(A) = (eQ(1−e), (1−e)Qe). Thus A ⊆ Q ⊆ Q_{max}^{l}(A), which obviously implies that Q is a left quotient pair of A. Now, AQ = AAQA, because A is a Fountain-Gould left order in AQ [for any x ∈ AQ, x = a^{#}b, where a ∈ A, a^{#} ∈ AQ, b ∈ A, hence x = a(a^{#})^{2}b ∈ AAQA], hence, Q^{σ} = A^{σ}Q^{−σ}A^{σ} [for example Q^{+} = eAQ(1 − e) = eAAQA(1 − e) = e(A^{+} + A^{+}A^{−})AQ(A^{−}A^{+} + A^{+})(1 − e) ⊆ eA^{+}AQA^{+}(1 − e) = A^{+}AQA^{+} = A^{+}(1 − e)AQeA^{+} = A^{+}Q^{−}A^{+}]. Now, (b) follows from (3.7).

(iii)⇔(iv) follows from [8, 5.2(iv), 5.3].

(v)⇒(iii) is just [8, 8.8(i)] together with (3.6).

(i)⇒(v) follows from (2.11) and (3.5).

Finally (1) and (2) are consequences of [8, 8.8], (2.11), and (3.6), since A is left local Goldie.

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