Left Quotient Associative Pairs and Morita Invariant Properties

Introduction

Along the years, the study of the Morita invariance has been a present question. In this paper, we study the Morita-invariance of properties closely related to the rings of quotients such as left nonsingularity and left nonsingularity plus finite left local Goldie dimension for idempotent rings without total left or right zero divisors (Sec. 4). We will get these results ((4.3) and (4.7)) as an application of the theory of associative pairs of quotients we develop.

Associative pairs and Morita contexts are closely related since every Morita context (ℛ, 𝒮, M, N) gives rise to an associative pair: (M, N) and, conversely, given an associative pair A = (A ⁺, A ⁻), there exists a unital ring ℰ with an idempotent e such that if ℰ = eℰe ⊕ (1 − e)ℰ(1 − e) ⊕ eℰ(1 − e) ⊕ (1 − e)ℰe is the Peirce decomposition of ℰ relative to e, then A is isomorphic to (eℰ(1 − e), (1 − e)ℰe) (and, of course, (eℰe, (1 − e)ℰ(1 − e), eℰ(1 − e), (1 − e)ℰe) is a Morita context). This ring ℰ is called the standard imbedding of A. Associative pairs play also a fundamental role in the new approach to Zelmanov's classification of strongly prime Jordan pairs (see D'Amour, Citation1992), and have been already used by Loos in the classification of the nondegenerate Jordan pairs of finite capacity (see Loos, Citation1975).

After a preliminary paragraph on associative pairs, in the first section we study the left singular ideal of an associative pair: we prove that, in general, it is a pair of two-sided ideals, and an ideal when the associative pair has no total right zero divisors (this is the case, for example, if the pair is semiprime). Nonsingularity of an associative pair A (i.e., Z _l (A) = 0, where Z _l (A) denotes the left singular ideal of A) is equivalent to that of its standard imbedding.

In Sec. 2, we introduce the notion of left quotient pair of a pair, extending the well-known definition of left quotient ring given by Utumi (Citation1956). We connect properties of a pair with those of its left quotient pairs, and find the maximal left quotient pair of every associative pair which has neither total left nor total right zero divisors, or it is left nonsingular.

Section 3 is devoted to Johnson and Gabriel's Theorems for associative pairs (see Johnson's Theorem for pairs and (3.3)). As a corollary we obtain the Gabriel's Theorem for associative pairs.

0. Preliminaries

An associative pair over a unital commutative associative ring of scalars Φ is a couple (A ⁺, A ⁻) of Φ-modules together with Φ-trilinear maps

By Loos (Citation1995, (2.3)) given an associative pair A = (A ⁺, A ⁻), there exists a unital associative algebra ℰ with an idempotent e such that if ℰ₁₁ ⊕ ℰ₁₂ ⊕ ℰ₂₁ ⊕ ℰ₂₂ is the Peirce decomposition of ℰ relative to e, that is, ℰ _ij  = u _i ℰu _j , with u ₁ = e and u ₂ = 1 − e, then A is isomorphic to the associative pair (ℰ₁₂, ℰ₂₁), where ℰ₁₁ (resp. ℰ₂₂) is spanned by e and all products x ₁₂ y ₂₁ (resp. 1 − e and all products y ₂₁ x ₁₂) for x ₁₂ ∈ ℰ₁₂, y ₂₁ ∈ ℰ₂₁, and has the following property:

The pair (ℰ, e) is called the standard imbedding of A.

Now let 𝒜 be the subalgebra of ℰ generated by the elements x ₁₂ and x ₂₁. It is immediate that 𝒜 is an ideal of ℰ. We will call 𝒜 the envelope of the associative pair A. Notice that 𝒜₁₂ ≔ e𝒜(1 − e) = ℰ₁₂ and 𝒜₂₁ ≔ (1 − e)𝒜e = ℰ₂₁, hence the associative pair A is isomorphic to the associative pair (𝒜₁₂, 𝒜₂₁). It is not difficult to see, by using (Equation0.2), that 𝒜 is an essential ideal of ℰ.

In what follows, an expression of the type xy, with x ∈ A ^σ, y ∈ A ^−σ, must be understood by considering the associative pair A inside its envelope 𝒜. And xy = 0 means λ _A (x, y) = 0 and ρ _A (x, y) = 0, where λ(x, y)z = ρ(y, z)x = xyz.

1. The Left Singular Ideal of an Associative Pair

The notion of left singular ideal for semiprime associative pairs was introduced by the authors jointly with Fernández López and García Rus in Fernández López et al. (Citation1998), where it was used as a tool to study Fountain–Gould orders in associative pairs. Here we analyze some properties of this left singular ideal without the semiprimeness hypothesis.

We recall that a nonzero element a in a ring ℛ is said to be a total right zero divisor if ℛa = 0.

1.1. Definitions

Let A be an associative pair. We will say that an element a ∈ A ^σ is a total right zero divisor if a is nonzero and A ^σ A ^−σ a = 0. When A ⁺ and A ⁻ have no total right zero divisors we will say that the associative pair A has no total right zero divisors.

These definitions are consistent with the classical ones because a right zero divisor a in a ring ℛ is a right zero divisor in the associative pair (ℛ, ℛ). Moreover ℛ has no total right zero divisors if and only if (ℛ, ℛ) has no total right zero divisors.

If A is an associative pair which has no total right zero divisors then, for every subset X of A ^σ,

Indeed, if b ∈ A ^−σ satisfies A ^σ bx = 0 for every x ∈ X, then bx A ^−σ must be zero. Otherwise, suppose bxd ≠ 0 for some d ∈ A ^−σ. Since A has no total right zero divisors, then there exist u ∈ A ^σ, v ∈ A ^−σ satisfying 0 ≠ vubxd ∈ v A ^σ bx A ^−σ = 0, a contradiction.

If moreover A has no total left zero divisors (for example, if it is a semiprime pair),

1.4. Lemma

If an associative pair A has no total right zero divisors, then (Equation0.2) is equivalent to:

Proof

Clearly, this conditions imply (Equation0.2). Conversely, 𝒜₂₁ x ₁₁ = 0 and x ₁₁ ≠ 0 implies, by (Equation0.2), x ₁₁ a ₁₂ ≠ 0 for some a ₁₂ ∈ 𝒜₁₂. Since A has no total right zero divisors, then 0 ≠ 𝒜₁₂𝒜₂₁ x ₁₁ a ₁₂, which is a contradiction.

A left ideal ℒ of a ring ℛ is dense in ℛ if for every x, y ∈ ℛ, with x ≠ 0, there exists an element a ∈ ℛ such that ay ∈ ℒ and ax ≠ 0. By Utumi (Citation1956, (1.6)), this condition is equivalent to say that ℛ is a left quotient ring of ℒ.

1.5. Lemma

Let A be an associative pair and write 𝒜 and ℰ to denote the envelope and the standard imbedding, respectively, of A. Then, the following conditions are equivalent:

Proof

(i) ⇒ (iii) Let x and y be elements in ℰ with x ≠ 0. Suppose first that x ₁₂ ≠ 0. By the hypothesis there exists (a ₁₂, a ₂₁) ∈ A such that a ₁₂ a ₂₁ x ₁₂ ≠ 0. Then a ₂₁ x ≠ 0 (otherwise 0 = a ₂₁ x = a ₂₁ x ₁₁ + a ₂₁ x ₁₂ implies a ₂₁ x ₁₁ = a ₂₁ x ₁₂ = 0, a contradiction) and a ₂₁ y ∈ 𝒜 since 𝒜 is an ideal of ℰ. The case x ₂₁ ≠ 0 is analogue. Now, suppose x ₁₂ = x ₂₁ = 0. In this case x ₁₁ (or x ₂₂) must be nonzero and by (1.4), b ₂₁ x ₁₁ ≠ 0 for some b ₂₁ ∈ 𝒜₂₁. Then, as it is easy to show, b ₂₁ x ≠ 0 and b ₂₁ y ∈ 𝒜.

(iii) ⇒ (ii) follows by Utumi (Citation1956, (1.6)).

(ii) ⇒ (i) Take 0 ≠ a ₁₂ ∈ 𝒜₁₂. By the hypothesis, there exist b, c ∈ 𝒜 such that bc a ₁₂ ≠ 0. This implies 𝒜₁₂𝒜₂₁ a ₁₂ ≠ 0.

For an associative pair A, it is defined

The “moreover” part of the following lemma can be obtained as a corollary of (1.9). However we include here a direct proof of the result.

1.6. Lemma

For an associative pair A we have that Z _l (A) = (Z _l (A)⁺, Z _l (A)⁻) is a pair of two-sided ideals of A. Moreover, if A has no total right zero divisors, then Z _l (A) is an ideal of A. In particular, Z _l (A) is an ideal of A if A is a semiprime associative pair.

Proof

Being Z _l (A) a pair of two-sided ideals of A follows the ideas of the proof of Theorem 3.1 in Fernández López et al. (Citation1998). Now, suppose that A has no total right zero divisors. Let x, y ∈ A ^σ and z ∈ Z _l (A)^−σ, and take a nonzero element l in a nonzero left ideal L of A contained in A ^−σ. If A ^σ lx = 0, then A ^−σ A ^σ lxzy = 0 and since A has no total right zero divisors, this implies 0 ≠ A ^−σ A ^σ l ⊆ L ∩ lan(xzy). If A ^σ lx ≠ 0, since lan(z) is an essential left ideal of A, there exists 0 ≠ alx ∈ A ^σ lx ∩ lan(z). Apply that A has no total right zero divisors to find b ∈ A ^σ, c ∈ A ^−σ such that bcalx ≠ 0. Then 0 ≠ cal ∈ L ∩ lan(xzy). In any case L ∩ lan(xzy) ≠ 0, so lan(xzy) is essential, which completes the proof.

1.7. Definitions

Given an associative pair A, the pair Z _l (A) = (Z _l (A)⁺, Z _l (A)⁻) of two-sided ideals of A will be called the left singular(two − sided)ideal of A. An associative pair A = (A ⁺, A ⁻) will be called left nonsingular if its left singular ideal Z _l (A) is zero. Right nonsingular associative pairs are defined similarly, while nonsingular means that A is both left and right nonsingular.

1.8. Lemma

Let A be a left nonsingular associative pair. Then A has no total right zero divisors.

Proof

We prove first the following property:

(1) For every nonzero x _ii  ∈ 𝒜 _ii we have 𝒜 _ji x _ii  ≠ 0 (for i ≠ j).

Suppose 𝒜 _ji x _ii  = 0 for some nonzero x _ii  ∈ 𝒜 _ii . Then 𝒜 _ji  = la n _A (x _ii 𝒜 _ij ). Since x _ii  ≠ 0, by (Equation0.2), x _ii 𝒜 _ij  ≠ 0 and we have just proved that x _ii 𝒜 _ij  ⊆ Z _l (A)^σ, for σ = + or σ = − , contrary to the hypothesis.

Now, let a ₁₂ be an element of 𝒜₁₂ such that 𝒜₁₂𝒜₂₁ a ₁₂ = 0; then, by (1), 𝒜₂₁ = la n _A (a ₁₂), which implies a ₁₂ ∈ Z _l (A)⁺ = 0 and proves that A has no total right zero divisors.

1.9. Proposition

Let A be an associative pair without total right zero divisors and denote by 𝒜 and ℰ its envelope and standard imbedding, respectively. Then

Proof

(1) Z _l (ℰ) ∩ A ^σ ⊆ Z _l (A)^σ.

Suppose x ∈ Z _l (ℰ) ∩ 𝒜₁₂ and take a nonzero left ideal L of A contained in 𝒜₂₁. If Lx = 0 then L ⊆ la n _A (x). If l ₂₁ x ≠ 0 for some l ₂₁ ∈ L, since ℰl ₂₁ is a nonzero left ideal of ℰ and la n _ℰ(x) is an essential left ideal of ℰ, there exists 0 ≠ u l ₂₁ ∈ ℰl ₂₁ ∩ la n _ℰ(x). Write u l ₂₁ = u ₁₂ l ₂₁ + u ₂₂ l ₂₁. If u ₂₂ l ₂₁ ≠ 0, then we have a nonzero element in L ∩ la n _A (x). If u ₁₂ l ₂₁ ≠ 0, by (1.4), 0 ≠ 𝒜₂₁ u ₁₂ l ₂₁ ⊆ L ∩ la n _A (x). Anyway, la n _A (x) is an essential left ideal of A contained in A ⁻, which implies x ∈ Z _l (A)⁺.

(2) Z _l (A)^σ ⊆ Z _l (𝒜) ∩ A ^σ.

Let x be in Z _l (A)⁺. Take a nonzero left ideal ℒ of 𝒜. If ℒx = 0 then ℒ ⊆ la n _𝒜(x). Suppose ℒx ≠ 0 and take l ∈ ℒ such that lx ≠ 0. If l ₂₁ ≠ 0, then ℰ₂₂ l ₂₁ is a nonzero left ideal of A contained in A ⁻; applying x ∈ Z _l (A)⁺ we find u ₂₂ ∈ ℰ₂₂ such that 0 ≠ u ₂₂ l ₂₁ ∈ la n _A (x) and this implies u ₂₂ lx = u ₂₂ l ₂₁ x = 0, so 0 ≠ u ₂₂ l ∈ ℒ ∩ la n _𝒜(x). If l ₁₁ ≠ 0, by (1.5), 𝒜l ₁₁ ≠ 0, so there exists a ₂₁ ∈ 𝒜₂₁ such that 0 ≠ a ₂₁ l ₁₁, and the element a ₂₁ l satisfies the conditions of the previous case.

(3) Z _l (𝒜) = Z _l (ℰ) ∩ 𝒜.

This follows by Proposition 3.2(ii) of Gómez Lozano and Siles Molina (Citation2002) since by (1.5), 𝒜 is a dense ideal of ℰ.

By (1), (2) and (3), Z _l (A)^σ ⊆ Z _l (𝒜) ∩ A ^σ = Z _l (ℰ) ∩ A ^σ ⊆ Z _l (A)^σ and the first statement has been proved.

(i) ⇒ (iii) Consider x ∈ Z _l (ℰ). Then x ₁₂ ∈ Z _l (ℰ) ∩ A ⁺ = Z _l (A)⁺ = 0 and analogously x ₂₁ = 0. If x _ii  ≠ 0, by (1.8) and (1.4), 0 ≠ 𝒜 _ji x _ii  ⊆ Z _l (ℰ) ∩ 𝒜 _ji  = Z _l (A)^σ = 0, which is a contradiction.

(ii) ⇔ (iii) follows by Proposition 3.2 (iii) of Gómez Lozano and Siles Molina (Citation2002), taking into account (1.5).

(ii) ⇒ (i) follows since Z _l (A)^σ = Z _l (𝒜) ∩ A ^σ.

1.10. Proposition

Let A be an associative pair without total right zero divisors and denote by 𝒜 and ℰ its envelope and standard imbedding, respectively. Then

• ( iii) 𝒜₁₁ is left nonsingular if and only if 𝒜₂₂ is left nonsingular.

Proof

Z _l (ℰ) ∩ 𝒜₁₁ ⊆ Z _l (𝒜₁₁): Take x ₁₁ ∈ Z _l (ℰ) ∩ 𝒜₁₁. We will see that lan _{𝒜 11} (x ₁₁) is an essential left ideal of 𝒜 ₁₁. Consider a nonzero element y in an ideal I of 𝒜 ₁₁. Then ℰy is a nonzero ideal of ℰ and there exists 0 ≠ uy ∈ la n _ℰ(x ₁₁) ∩ ℰy. If 0 ≠ euy ∈ lan_𝒜11 (x ₁₁)∩I we have finished. Otherwise, 0 ≠ uy = (1 − e)uy and since A has no total right zero divisors, 0 ≠ 𝒜 ₁₂ uy ⊆ lan _{𝒜 11} (x ₁₁) ∩ I.

Z _l (𝒜₁₁) ⊆ Z _l (𝒜) ∩ 𝒜₁₁: Consider x ∈ Z _l (𝒜₁₁). We want to prove that lan _𝒜(x ₁₁) is an essential left ideal of 𝒜. Take a nonzero left ideal I of 𝒜. If Ie = 0 then I ⊆ la n _𝒜(x ₁₁). If Ie ≠ 0 we have eIe ≠ 0 (suppose (1 − e)Ie ≠ 0; apply that A has no total right zero divisors to obtain 0 ≠ 𝒜 ₁₂Ie ∩ lan _{𝒜 11} (x ₁₁), which implies 0 ≠ 𝒜 ₁₂ I ∩ lan_𝒜11 (x ₁₁) ⊆ I ∩ lan_𝒜 (x ₁₁)) and hence 0 ≠ eIe ∩ lan_𝒜11 (x ₁₁), which implies 0 ≠ eI ∩ la n _𝒜(x ₁₁) ⊆ I ∩ la n _𝒜(x ₁₁).

Now, notice that Z _l (𝒜 _ii ) ⊆ Z _l (𝒜) ∩ 𝒜 _ii  = (by Gómez Lozano and Siles Molina (2002, 3.2(ii)) in the proof of (1.9)) Z _l (ℰ) ∩ 𝒜 _ii  ⊆ Z _l (𝒜 _ii ), which proves the first statement.

(i) ⇔ (ii) If ℰ is left nonsingular, by (1.9), A is left nonsingular. Then Z _l (𝒜₁₁) = Z _l (𝒜₂₂) = 0. Conversely, suppose Z _l (𝒜 _ii ) = 0 for i, j = 1, 2, i ≠ j. Since Z _l (ℰ) is an ideal, x _ij  ∈ Z _l (ℰ), and by the first statement, x _ii  ∈ Z _l (𝒜 _ii ), for i = 1, 2. Now, since A has no total right zero divisors, if x _ij  ≠ 0, 𝒜 _ji x _ij  ≠ 0. Hence 0 ≠ 𝒜 _ji x _ij  ≠ Z _l (ℰ) ∩ 𝒜 _jj  = Z _l (𝒜 _jj ) = 0, a contradiction.

(iii) Suppose 𝒜₁₁ left nonsingular, and consider 0 ≠ x ₂₂ ∈ Z _l (𝒜₂₂). By (1.4) 𝒜₁₂ x ₂₂ ≠ 0 and since A has no total left zero divisors, 0 ≠ 𝒜₁₂ x ₂₂𝒜₂₁𝒜₁₂. This means 0 ≠ 𝒜₁₂ x ₂₂𝒜₂₁ ⊆ 𝒜₁₂ Z _l (𝒜₂₂)𝒜₂₁ ⊆ (by the previous statement) Z _l (ℰ) ∩ 𝒜₁₁ = Z _l (𝒜₁₁) = 0, which is not possible. Hence Z _l (𝒜₂₂) = 0.

2. Left Quotient Pairs

The notion of left quotient ring was introduced by Utumi (Citation1956) and has proved to be very useful in order to study Fountain–Gould left orders in rings (see Gómez Lozano and Siles Molina, Citation2002 and the related references therein). Let ℛ be a subring of a ring 𝒬. We say that 𝒬 is a (general) left quotient ring of ℛ if for every x, y ∈ 𝒬, with x ≠ 0, there is an a ∈ ℛ such that ax ≠ 0 and ay ∈ ℛ. Notice that a ring is a left quotient ring of itself if and only if it has no total right zero divisors. If ℛ has no total right zero divisors, then by Utumi (Citation1956) it has a unique maximal left quotient ring, which is unital, called the Utumi left quotient ring of ℛ.

2.1. Definition

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 (and σ = + or σ =−) there exist a ∈ A ^σ, b ∈ A ^−σ such that abp ≠ 0 and abq ∈ A ^σ.

For example, the associative pair (ℳ_1×2(ℚ), ℳ_2×1(ℚ)) is a left quotient pair of the associative pair (ℳ_1×2(4ℤ), ℳ_2×1(8ℤ)). Moreover, it is maximal among the left quotient pairs of (ℳ_1×2(4ℤ), ℳ_2×1(8ℤ)). Every associative pair without total right zero divisors is a left quotient pair of itself.

The notion of left quotient pair extends that of Utumi of left quotient ring since given a subring ℛ of an associative ring 𝒬, 𝒬 is a left quotient ring of ℛ if and only if Q = (𝒬, 𝒬) is a left quotient pair of the associative pair R = (ℛ, ℛ).

The following lemma will be used in the sequel although without mentioning it.

2.2. Lemma

Let Q = (Q ⁺, Q ⁻) be a left quotient pair of an associative pair A = (A ⁺, A ⁻). Then, given q ₁,…, q _n  ∈ Q ^σ with q ₁ ≠ 0 (σ = + or σ =−), there exist a ∈ A ^σ, b ∈ A ^−σ such that ab q ₁ ≠ 0 and ab q _i  ∈ A ^σ for all i ∈ {1,…, n}.

Proof

The case n = 1 follows from the definition. Suppose the result is true for n − 1. By the induction assumption there exist x ∈ A ^σ, y ∈ A ^−σ such that xy q ₁ ≠ 0 and xy q _i  ∈ A ^σ for i ∈ {1,…, n − 1}. Given xy q ₁ ≠ 0 and xy q _n , there exist z ∈ A ^σ, t ∈ A ^−σ such that ztxy q ₁ ≠ 0 and ztxy q _n  ∈ A ^σ. Take a = ztx ∈ A ^σ and b = y ∈ A ^−σ to complete the proof.

2.3

Let Q be an associative pair which is a left quotient pair of a subpair A. Then, it is not difficult to see that for any finite family in (Q ^σ, Q ^−σ), we have that is an essential left ideal of A.

2.4. Lemma

Let Q be an associative pair which is a left quotient pair of a subpair A. If A has no total left zero divisors (for example, if it is semiprime) or it is left nonsingular, then for any finite family in (Q ^σ, Q ^−σ) and every nonzero a ∈ A ^−σ we have:

Proof

Suppose first A without total left zero divisors. Then, given the nonzero element a, and applying that A has no total right zero divisors, there exist b, c ∈ A ^σ satisfying cab ≠ 0. Apply that Q is a left quotient pair of A to find x ∈ A ^σ, y ∈ A ^−σ such that xycab ≠ 0 and . Since the element xyc is in (A ^σ: and xyca ≠ 0, we have finished.

Now, suppose A left nonsingular. If , then (A ^σ: , which implies, by (2.3), 0 ≠ a ∈ Z _l (A)^−σ, a contradiction.

2.5. Proposition

Let A be a subpair of an associative pair Q, and write 𝒜 and 𝒬 to denote the envelopes of A and Q, respectively.

• (iii) If Q is a left quotient pair of A, then 𝒬 is a left quotient ring of 𝒜.

Proof

(i) By the construction of ℰ, ℰ₁₁ is the subalgebra of En d _Φ(A) × En d _Φ(A ⁻) ^op generated by {(λ _A (x, y), ρ _A (x, y)) | x ∈ A ⁺, y ∈ A ⁻} and (Id _{A ⁺} , Id _{A ⁻} ), where the index A, A ⁺ or A ⁻ under each operator means where it acts. Analogously we have that ℰ₁₁′ is the subalgebra of En d _Φ(Q ⁺) × , En d _Φ(Q ⁻) ^op generated by {(λ _Q (x, y), ρ _Q (x, y)) | x ∈ Q ⁺, y ∈ Q ⁻} and (Id _{Q ⁺} , Id _{Q ⁻} ). Hence, to prove ℰ ⊆ ℰ′, and since A ⁺ = ℰ₁₂ ⊆ ℰ₁₂′ = Q ⁺, it is enough to show that for every n ∈ ℕ ∪ {0}, x _i  ∈ A ⁺, y _i  ∈ A ⁻, the map from ℰ₁₁ into ℰ₁₁′ which sends n(Id _{A ⁺} , Id _{A ⁻} ) +  )∑ λ _A (x _i , y _i ), ∑ ρ_A x _i , y _i )) to n(Id _{Q ⁺} , Id _{Q ⁻} ) +  )∑ λ _Q (x _i , y _i ), ∑ ρ_Q x _i , y _i )) is well-defined. Or, equivalently, that

Notice that with this reasoning we have:

2.6

If p _ii is a nonzero element of 𝒬 _ii , then 𝒜 _ji p _ii  ≠ 0, for i ≠ j, i, j ∈ {1, 2}.

Now we will see e = e′. Since 𝒜₂₁(e − e′) = 0, by (2.6), e = e′. Analogously we can prove 1_ℰ − e = 1_ℰ′ − e′, which leads to 1_ℰ = 1_ℰ′.

(ii) Take p ₁₂, q ₁₂ ∈ 𝒬₁₂ with p ₁₂ ≠ 0. By the hypothesis there exists b ∈ 𝒜 such that b p ₁₂ ≠ 0 and b p ₁₂, b q ₁₂ ∈ 𝒜. This implies b ₁₁ p ₁₂ ≠ 0 or b ₂₁ p ₁₂ ≠ 0. Suppose first 0 ≠ b ₁₁ p ₁₂ ∈ 𝒜₁₂. By (1.5), A has no total right zero divisors, hence there exists (c ₁₂, c ₂₁) ∈ A satisfying c ₁₂ c ₂₁ b ₁₁ p ₁₂ ≠ 0. If we denote d ₂₁ = c ₂₁ b ₁₁, then c ₁₂ d ₂₁ p ₁₂ ≠ 0 and c ₁₂ d ₂₁ q ₁₂ ∈ 𝒜₁₂.

Now, consider 0 ≠ b ₂₁ p ₁₂ ∈ 𝒜₂₂. By (1.5) and (1.4), there exists c ₁₂ ∈ 𝒜₁₂ such that c ₁₂ b ₂₁ p ₁₂ ≠ 0. Moreover c ₁₂ b ₂₁ q ₁₂ = c ₁₂ b q ₁₂ ∈ 𝒜₁₂.

(iii) Let p and q be in 𝒬 with p ≠ 0. We distinguish two cases. Suppose first p ₁₁ ≠ 0 (the case p ₂₂ ≠ 0 is analogue). By (2.6) there exists a ₂₁ ∈ 𝒜₂₁ such that a ₂₁ p ₁₁ ≠ 0. Apply the hypothesis to find b ₁₂ ∈ 𝒜₁₂, b ₂₁ ∈ 𝒜₂₁ satisfying b ₂₁ b ₁₂ a ₂₁ p ₁₁ ≠ 0 and b ₂₁ b ₁₂ a ₂₁ p ₁₁, b ₂₁ b ₁₂ a ₂₁ q ₁₁ ∈ 𝒜₂₁. Write c ₂₁ = b ₂₁ b ₁₂ a ₂₁. By (2.4), (𝒜₁₂:c ₂₁ q ₁₂)c ₂₁ p ₁₁ ≠ 0. Let c ₁₂ be in (𝒜₁₂:c ₂₁ q ₁₂) satisfying c ₁₂ c ₂₁ p ₁₁ ≠ 0. Then the element d = c ₁₂ c ₂₁ ∈ 𝒜 verifies: dp = d p ₁₁ + d p ₁₂ ≠ 0 (since d p ₁₁ ≠ 0) and dq ∈ 𝒜.

Now, suppose p ₁₁ = p ₂₂ = 0. In this case p ₁₂ or p ₂₁ must be nonzero. Consider, for example, p ₁₂ ≠ 0. Apply that Q is a left quotient pair of A to find a ₂₁ ∈ 𝒜₂₁ such that a ₂₁ p ₁₂ ≠ 0. Then, given a ₂₁ p ₁₂ and a ₂₁ q, by the previous case there exists b ∈ 𝒜 such that b a ₂₁ p = b a ₂₁ p ₁₂ ≠ 0 and b a ₂₁ q ∈ 𝒜, which concludes the proof.

2.7. Example

There exist two associative pairs A and Q such that A ⊆ Q but there is no ring monomorphisms f:𝒜 → 𝒬 such that f(a) = a for every a ∈ A ^σ, where 𝒜 and 𝒬 denote the envelopes of A and Q, respectively.

Proof

Consider a ring ℛ such that ℛ⁵ ≠ 0 but ℛ⁶ = 0 and define A = (ℛ², ℛ²), Q = (ℛ, ℛ). It is clear that A ⊆ Q. If f:𝒜 → 𝒬 is a ring monomorphism such that f(a) = a for every a ∈ A ^σ, then λ _A (a, b) = 0 implies λ _Q (a, b) = 0, with (a, b) ∈ A. But this is not the case: Choose a, x, y, z, t ∈ ℛ such that axyzt ≠ 0 and define b = xyz. Then λ _A (a, b)A ^σ = abℛ² ⊆ ℛ⁶ = 0, and 0 ≠ abt ∈ λ _Q (a, b)Q ^σ.

2.8. Lemma

If Q is a left quotient pair of a subpair A, and (ℰ′, e′) and (ℰ, e′) are the standard imbeddings of Q and A, respectively, then ℰ₁₁′ ≔ e′ℰ′e′ is a left quotient ring of 𝒜₁₁ ≔ e′𝒜e′, where 𝒜 denotes the envelope of A.

Proof

By (2.5) (i), ℰ ⊆ ℰ′, so 𝒜₁₁ ⊆ ℰ₁₁′. Consider p, q ∈ ℰ₁₁′, with p ≠ 0. By (2.6), a ₂₁ p ≠ 0 for some a ₂₁ ∈ 𝒜₂₁. By the hypothesis there exist b ₁₂ ∈ 𝒜₁₂, c ₂₁ ∈ 𝒜₂₁ such that c ₂₁ b ₁₂ a ₂₁ p ≠ 0 and c ₂₁ b ₁₂ a ₂₁ p, c ₂₁ b ₁₂ a ₂₁ q ∈ 𝒜₂₁. Since A has no total right zero divisors, there exists d ₁₂ ∈ 𝒜₁₂ such that d ₁₂ c ₂₁ b ₁₂ a ₂₁ p ≠ 0. If we denote x = d ₁₂ c ₂₁ b ₁₂ a ₂₁ ∈ 𝒜₁₁, then xp ≠ 0 and xq ∈ 𝒜₁₁.

2.9. Theorem

Let A be an associative pair, and denote by 𝒜 and (ℰ, e) its envelope and standard imbedding, respectively.

Proof

(i) By (1.5), 𝒜 is a dense ideal of ℰ and by Exercise 13.21 of Lam (Citation1999), . (ii) First of all we are going to see:

2.10

For every 0 ≠ x ∈ 𝒜, eℰx ≠ 0 and (1 − e)ℰx ≠ 0.

Indeed, let x be in 𝒜 such that eℰx = 0. Then 0 = ex = exe + ex(1 − e), which implies x ₁₁ = x ₁₂ = 0. Now, 𝒜₁₂ x ⊆ eℰx = 0 implies 𝒜₁₂ x ₂₁ = 𝒜₁₂ x ₂₂ = 0; apply that A is a left quotient pair of A and (2.6) to obtain x ₂₁ = x ₂₂ = 0 and, hence x = 0.

Now, let p and q be in e𝒬(1 − e) with p ≠ 0. Apply that 𝒬 is a left quotient ring of 𝒜 to take a ∈ 𝒜 such that 0 ≠ ap ∈ 𝒜 and aq ∈ 𝒜. By (2.10) applied twice, there exist x, y ∈ ℰ such that ey(1 − e)xaep = ey(1 − e)xap ≠ 0. Moreover the elements ey(1 − e) ∈ 𝒜₁₂ and (1 − e)xae ∈ 𝒜₂₁ satisfy ey(1 − e)xaeq = ey(1 − e)xaq ∈ 𝒜₁₂.

Finally, suppose that T is a left quotient pair of A and write (ℰ _T , e) to denote the standard imbedding of T. By (2.5) (iii), ℰ _T is a left quotient ring of 𝒜 and therefore 𝒜 ⊆ 𝒯 ⊆ 𝒬. So, (e𝒯(1 − e), (1 − e)𝒯 e) ⊆ (e𝒬(1 − e), (1 − e)𝒬e).

2.11. Definition

If A is an associative pair without total left zero divisors and without total right zero divisors, or it is left nonsingular, then by (2.9)(ii), it has a unique (up to isomorphisms) maximal left quotient associative pair which will be called the maximal left quotient pair of A and if we denote it by , then (by 2.9(ii)),

A natural question which arises is if the envelope of the maximal left quotient pair of an associative pair A and the maximal left quotient ring of the envelope coincide. The answer is negative, as it is shown with the following example.

2.12. Example

Let V be a left vector space over a field K of infinite dimension, 𝒬 = En d _K (V) and 𝒜 = Soc(𝒬). Consider two idempotents e, f ∈ 𝒬 such that e + f = 1, e ∈ 𝒜, f ∉ 𝒜. Then the associative pair A = (e𝒜f, f𝒜e) has the ring 𝒜 as an envelope, , and the envelope of is e𝒬f ⊕ f𝒬e ⊕ e𝒬f𝒬e ⊕ f𝒬e𝒬f = 𝒜 ≠ 𝒬.

2.13. Lemma

If Q is a left quotient pair of an associative pair A, then:

Proof

(i) follows from the definition.

(ii) Suppose A prime. If I and J are two nonzero left ideals of Q, by (i), I ^σ ∩ A ^σ and J ^σ ∩ A ^σ are nonzero, hence (I ^σ ∩ A ^σ)A ^−σ(J ^σ ∩ A ^σ) ≠ 0, which implies I ^σ Q ^−σ J ^σ ≠ 0. The case A semiprime follows analogously by considering J = I.

With the following proposition we show the relationship between some properties of an associative pair and the analogues of its left quotient pairs (see Fernández López et al., Citation1998 for the definitions).

We recall the notion of local ring at an element of an associative pair (see Fernández López et al., Citation1998): Let A = (A ⁺, A ⁻) be an associative pair and a ∈ A ^σ. Then the submodule a A ^−σ a equipped with the multiplication defined by (axa) ⋅ (aya) = axaya is a ring called the local ring of A at a and denoted by A _a . Note that if a is von Neumann regular, i.e., a ∈ a A ^−σ a, then A _a is unital with a as the unity.

A family of left ideals {L _i } _i∈Γ of an associative pair is said to be independent if the sum of its ideals is direct.

2.14. Proposition

Let Q be a left quotient pair of an associative pair A. Then:

Proof

(i) It is clear that for any X ⊆ A ⁺, la n _Q (X) ∩ A ⁻ ⊆ la n _A (X). Conversely, let z be in la n _A (X). Then, for every x ∈ X, 0 = zx ∈ 𝒜₂₂ ⊆ 𝒬₂₂ (by (2.5) (i)). Therefore, z ∈ la n _Q (X).

(ii) follows by (i).

(iii) Let z be in Z _l (A)⁺. We have to see that la n _Q (z) is an essential left ideal of Q. Let L be a nonzero left ideal of Q contained in Q ⁻. By (2.13)(i), L ∩ A ⁻ is a nonzero left ideal of A. Since la n _A (z) is an essential left ideal of A, la n _A (z) ∩ L ∩ A ⁻ ≠ 0, and by (i) la n _Q (z) ∩ L ≠ 0.

Conversely, suppose z ∈ Z _l (Q) ∩ A and let prove that la n _A (z) is an essential left ideal of A. Given a nonzero left ideal L of A contained in A ⁻, since A has no total right zero divisors we have A ⁻ A ⁺ L ≠ 0, which implies Q ⁻ al ≠ 0 for some a ∈ A ⁺, l ∈ L. Apply that la n _Q (z) is an essential left ideal of Q to find 0 ≠ pal ∈ Q ⁻ al ∩ la n _Q (z). Since Q is a left quotient pair of A, given 0 ≠ pal and p, there exist u ∈ A ⁺, v ∈ A ⁻ such that vupal ≠ 0 and vup ∈ A ⁺. Then 0 ≠ vupal ∈ A ⁺ A ⁻ L ∩ la n _Q (z) ⊆ L ∩ la n _A (z) (by (i)).

(iv) By (iii), Z _l (Q) = 0 implies Z _l (A) = 0. Conversely, if Z _l (A) = 0 then Z _l (Q) must be zero by (2.13)(i).

(v) Let {L _i } _i∈Γ be as in the statement. For every i ∈ Γ, choose a nonzero element l _i  ∈ L _i . Since A has no total right zero divisors, there exist a _i  ∈ A ^σ, b _i  ∈ A ^−σ such that 0 ≠ a _i b _i l _i . Then [Ltilde] _i  = Q ^σ b _i l _i is a nonzero left ideal of Q contained in Q ^σ. Now we see that the sum of the [Ltilde] _i 's is direct.

Suppose q ₁ b ₁ l ₁ = q ₂ b ₂ l ₂ + … + q _n b _n l _n for , with q ₁ b ₁ l ₁ ≠ 0. Apply that Q is a left quotient pair of A to find u ∈ A ^σ, v ∈ A ^−σ such that uv q ₁ b ₁ l ₁ ≠ 0 and uv q _i  ∈ A ^σ. Then , a contradiction.

(vi) follows immediately by taking into account (2.13)(i).

(vii) is a direct consequence of (v) and (vi).

(viii) follows by (v).

(ix) Take apa, aqa ∈ Q _a , with apa ≠ 0. Apply that Q is a left quotient pair of A to find x ∈ A ^σ, y ∈ A ^−σ satisfying 0 ≠ xyapa ∈ A ^σ. By (2.4), given xya ∈ Q ^σ, q ∈ Q ^−σ and 0 ≠ xyapa ∈ A ^σ, we have (A ^−σ:xyaq)xyapa ≠ 0. Let z be in A ^−σ with zxyaq ∈ A ^−σ and zxyapa ≠ 0. By (1.4) and the semiprimeness of A, 0 ≠ atuzxyapa for some t ∈ A ^−σ, u ∈ A ^σ. Then the element atuzxya, which is in A _a , satisfies: atuzxya.apa = atuzxyapa ≠ 0 and atuzxya.aqa = atuzxyaqa ∈ A _a .

(x) By (ii), la n _Q (X) ⊆ la n _Q (Y) implies la n _A (X) ⊆ la n _A (Y). Now we prove the converse. Suppose la n _A (X) ⊆ la n _A (Y) but la n _Q (X)¬ ⊆ la n _Q (Y) and let q be in la n _Q (X) such that pqy ≠ 0 for some y ∈ Y, p ∈ Q ^σ (this is possible by virtue of (1.4)). Since Q is a left quotient pair of A, there exist u ∈ A ^σ, v ∈ A ^−σ such that uvpqy ≠ 0 and uvpqy ∈ A ^σ. By (2.4), (A ^−σ:uvpq) uvpqy ≠ 0, so there exists b ∈ A ^−σ such that buvpq ∈ A ^−σ and 0 ≠ buvpqy. Then buvpq ∈ la n _A (X) (because q ∈ la n _Q (X)) but buvpq ∉ la n _A (Y), which contradicts the initial hypothesis.

(xi) By (viii), u-dim _A (a) ≤ u-dim _Q (a). Now, let {[Ltilde] _i } _i∈Γ be a family of nonzero left ideals of Q contained in Q ^σ Q ^−σ a. We can take , with p _{k i} , q _{k i}  ∈ Q ^σ. By (2.4), (A ^−σ : p _{k i} q _{k i} ) a ≠ 0, hence there exists y _i in A ^−σ such that ∑ y _i p _{k i} q _{k i}  ∈ A ^−σ and ∑ y _i p _{k i} q _{k i}  ∈ a ≠ 0. By (1.4) we can find x ∈ A ^σ such that l _i  ≔ xy _i ∑ p _{k i} q _{k i} a ≠ 0. Since l _i  ∈ A ^σ A ^−σ a ∩ [Ltilde] _i , {A ^σ A ^−σ l _i } _i∈Γ is a family of nonzero left ideals of A contained in A ^σ A ^−σ a. Moreover its sum is direct (because A ^σ A ^−σ l _i  ⊆ [Ltilde] _i and the sum of the [Ltilde] _i 's was direct), which proves our claim.

3. Johnson and Gabriel's Theorems for Associative Pairs

While Johnson's Theorem characterizes those rings ℛ for which is von Neumann regular (Lam, Citation1999, 13.36), Gabriel's Theorem (Lam, Citation1999, 13.40) specializes it further by asking for characterizations for those rings ℛ for which is semisimple, i.e., isomorphic to a finite direct product of rings of the form En d _Δ(V) for a suitable finite left vector space V over a division ring Δ. In this section we prove that every associative pair A for which is von Neumann regular is left nonsingular (and conversely), and characterize those associative pairs whose maximal left quotient pair is isomorphic to Π_α∈Γ (Hom_Δα (V _α, W _α), Hom_Δα (W _α, V _α)), where for each α ∈ Γ, V _α and W _α are left vector spaces over the same division ring Δ_α. In particular we get a characterization of those associative pairs whose maximal left quotient pair is semisimple and artinian.

3.1. Johnson's Theorem for Associative Pairs

Let A be an associative pair. Then A is left nonsingular if and only if is a von Neumann regular associative pair.

Proof

Suppose A is left nonsingular. By (1.9), ℰ is left nonsingular. By Johnson's Theorem for rings (Lam, Citation1999, 13.36), the maximal left quotient ring of ℰ, name it 𝒬, is von Neumann regular. By 2.9(ii), , and it is easy to prove that this is a von Neumann regular associative pair. Conversely, as every von Neumann regular associative pair is left nonsingular (by Proposition 3.4 of Fernández López et al., Citation1998), 2.14(iv) completes the proof.

Let ℛ be an element in an arbitrary ring ℛ. Recall that the local ring of ℛ at a is defined as the ring obtained from the abelian group (aℛa, +) by considering the product given by axa ⋅ aya = axaya. Denote it by ℛ _a . The reader is referred to Gómez Lozano and Siles Molina (Citation2002) to see the relation among some properties of a ring and the corresponding ones of its local rings at elements.

Now, we will introduce some notation. Given a ring ℛ and an element x ∈ ℛ, the left Goldie (or uniform) dimension of x will be denoted by u-dim(x) (u-dim_ℛ(x) to specify the ring). By u-dim(R) we understand the uniform dimension of _R R. We put I(ℛ) = {x ∈ ℛ|u − dim(x) < ∞}. Condition (iii) in the next Proposition was proved by Ánh and Márki (Citation1996, Proposition 1). Here we give a different proof by using Johnson's Theorem for rings.

3.2. Proposition

Let ℛ be a ring and denote by Soc(ℛ) the socle of the ring ℛ.

Proof

(i) The equality holds by [6, Proposition 2.1 (iv)] of Gómez Lozano and Siles Molina (Citation2002). Now, let x be in Soc(ℛ). Since Soc(ℛ) is a von Neumann regular ideal of ℛ, xℛx = xSoc(ℛ)x, which obviously implies u-dim(R _x ) = u-dim(Soc(ℛ) _x ). By Proposition 2.1 (vi) of Gómez Lozano and Siles Molina (Citation2002), u-dim(Soc(ℛ) _x ) <  ∞ and hence x ∈ I(ℛ).

(ii) By (i) we only need to prove I(ℛ) ⊆ Soc(ℛ). Take a nonzero x ∈ I(ℛ). By Proposition 2.1 (i), (iv) and (ix) of Gómez Lozano and Siles Molina (Citation2002), ℛ _x is a semiprime left Goldie ring. By the classical Goldie's Theorem, ℛ _x is a classical left order in a semisimple artinian ring 𝒯. Since ℛ is von Neumann regular, the ring ℛ _x is unital and von Neumann regular. Therefore, Reg(ℛ _x ) = Inv(ℛ _x ), where “Reg” and “Inv” denote the set of regular and invertible elements, respectively. Since 𝒯 is generated by ℛ _x and the inverses of the elements of Reg(ℛ _x ), we have 𝒯 = ℛ _x and 𝒯(= ℛ _x ) artinian implies, by Proposition 2.1 (v) of Gómez Lozano and Siles Molina (Citation2002), x ∈ Soc(ℛ).

(iii) By Johnson's Theorem (Lam, Citation1999, 13.36), is a von Neumann regular ring. Since, by (ii), I(𝒬) = Soc(𝒬) is an ideal of 𝒬, clearly I(𝒬) ∩ ℛ is an ideal of ℛ. We conclude the proof by applying Proposition 3.2 (iv) of Gómez Lozano and Siles Molina (Citation2002), which says I(ℛ) = I(𝒬) ∩ ℛ.

Given an associative pair A, denote by I(A)^σ the set of the elements of A ^σ having finite left Goldie dimension, and set I(A) = (I(A)⁺, I(A)⁻). We denote by Soc(A) the socle of A (the reader is referred to Loos, Citation1991 for the study of the socle of an associative pair).

3.3. Proposition

Let A be an associative pair and denote by 𝒜 and (ℰ, e) its envelope and standard imbedding, respectively.

Proof

(i) By Proposition 5.2 (iv) of Fernández López et al. (Citation1998), a ∈ I(A)^σ if and only if u-dim(A _a ) <  ∞, and so the first equality holds.

Since A semiprime implies ℰ and 𝒜 semiprime (by Proposition 4.2 of Fernández López et al., Citation1998, ℰ is semiprime, and 𝒜 is semiprime because it is an ideal of ℰ), by condition (i) of (3.2) and taking into account that a A ^−σ a = a𝒜a = aℰa, for every a ∈ A ^σ, we have I(A)^σ = I(𝒜) ∩ A ^σ = I(ℰ) ∩ A ^σ.

Now, take an element a ∈ Soc(A)^σ and let b be in A ^−σ satisfying aba = a (which is possible by virtue of Loos, Citation1989, Theorem 1). Then a A ^−σ a = aba A ^−σ aba ⊆ a Soc(A)^σ a (because Soc(A) is an ideal of A). Since, obviously, a Soc(A)^σ a ⊆ a A ^−σ a, we have Soc(A) _a  = A _a . Finally, apply Proposition 5.2(v) of Fernández López et al. (Citation1998) to infer that u-dim(Soc(A)) <  ∞. By Proposition 5.2(iv) of Fernández López et al. (Citation1998), u-dim(Soc(A) _a ) <  ∞ hence Soc(A) ⊆ I(A).

(ii) By Theorem 2.9, there exists and Q ≔ (𝒬₁₂, 𝒬₂₁) is a left quotient pair of A. Moreover, since 𝒜 is left nonsingular (by Proposition 1.9), 𝒬 is von Neumann regular (Johnson's Theorem). Take an element a ₁₂ ∈ A ⁺. Then

(iii) Let a be in I(A). By Proposition 5.2(i), (iv) and Proposition 5.5 of Fernández López et al. (Citation1998), A _a is a semiprime left Goldie ring. Now we follow the same reasoning as in the proof of condition (ii) in (3.2) (notice that A _a is von Neumann regular since A is so) to prove A _a artinian. By (Fernández López et al. (Citation1998), Proposition 5.2 (v)) this implies a ∈ Soc(A).

(iv) By (2.9) (ii) and (2.9), is a von Neumann regular associative pair, so we can apply (iii) to obtain I(Q)^σ = Soc(Q)^σ, which is an ideal of A. Since I(A)^σ = I(Q)^σ ∩ A ^σ, by (2.14) (xi), we have proved the required statement.

3.4. Theorem

For an associative pair A the following conditions are equivalent:

Proof

(i) ⇒ (ii) By Johnson's Theorem for associative pairs, is a von Neumann regular associative pair and by 2.9 (ii), Q = (e𝒬(1 − e), (1 − e)𝒬e), where and 𝒜 is the envelope of A. Denote by 𝒮 the subalgebra of 𝒬 generated by Q, that is, 𝒮 is the envelope of Q. It is not difficult to see that 𝒮 is an ideal of 𝒬. Moreover,

(1) 𝒬 is a von Neumann regular ring.

By 1.9, A left nonsingular implies 𝒜 left nonsingular, and by Johnson's Theorem for rings, 𝒬 is a von Neumann regular ring.

(2) I(𝒜) is dense in 𝒜.

For every a ∈ A ^σ, u − dim _A (a) = u − dim _Q (a) (by 2.14 (xi)). This implies I(A)^σ ⊆ I(Q)^σ. Since I(A)^σ is an essential left ideal of A, I(Q)^σ is an essential ideal of Q. By 3.2 (ii), I(Q)^σ = I(𝒬) ∩ Q ^σ. We claim that I(𝒬) is an essential ideal of 𝒬: Consider a nonzero ideal J of Q ^σ. By Proposition 4.1 (i) of Fernández López et al. (Citation1998) J ∩ Q ^σ is a nonzero ideal of Q ^σ. Hence 0 ≠ I(Q)^σ ∩ J ∩ Q ^σ = I(𝒬) ∩ J ∩ Q ^σ. This shows our claim.

Finally, being 𝒬 a left quotient ring of 𝒜 implies I(𝒜) = I(𝒬) ∩ 𝒜 is an essential ideal of 𝒜 and consequently it is a dense ideal of 𝒜 (apply that 𝒜 is left nonsingular).

(3) The conclusion.

By Theorem 3.24 of Lam (Citation1999), which can be applied taking into account (2) and that 𝒜 is left nonsingular, , where each 𝒬_α is an ideal of 𝒬 isomorphic (as a ring) to End_Δα (U _α) for a suitable left vector space U _α over some division ring Δ_α.

Define V _α ≔ U _α e _α, W _α = U _α(1 − e)_α, e _α = π_α(e), (1 − e)_α = π_α(1 − e). Then we have (e𝒬(1 − e), (1 − e)𝒬e) ≅ Π (Hom_Δα (V _α, W _α), Hom_Δα (W _α, V _α)).

(ii) ⇒ (i)Define U _α = V _α ⊕ W _α, 𝒬 = Π_α End_Δα (U _α), e = (e _α) and f = (f _α), where

Notice that finiteness of left Goldie dimension of A implies that the direct product in the previous theorem must be finite as well as the dimensions of the vector spaces involved.

3.5. Gabriel's Theorem for Associative Pairs

For an associative pair A the following conditions are equivalent:

4. Applications to Morita Contexts

Let R and S be two rings, _R N _S and _S M _R two bimodules and (− , −):N × M → R, [−, −]:M × N → S two maps. Then the following conditions are equivalent:

i.	is a ring with componentwise sum and product given by:
ii.	[−, −] is S-bilinear and R-balanced, (− , −) is R-bilinear and S-balanced and the following associativity conditions holds: [−, −] being S-bilinear and R-balanced and (− , −) being R-bilinear and S-balanced is equivalent to having bimodule maps ϕ:N ⊗  S M → R and ψ:M ⊗  R N → S, given by so that the associativity condition above reads

A Morita context is a sextuple (R, S, N, M, ϕ, ψ) satisfying the conditions given above. The associated ring is called the Morita ring of the context.

In classical Morita theory it is shown that two rings with identity R and S are Morita equivalent (i.e., R- and S-mod are equivalent categories) if and only if there exists a Morita context (R, S, N, M, ϕ, ψ), with ϕ and ψ surjective. The approach to Morita theory for rings without identity by means of Morita contexts appears in a number of papers (see Marín, Citation1998 and the references therein) in which many consequences are obtained from the existence of a Morita context for two rings R and S. In particular it is shown in Theorem of Kyuno (Citation1974) that, if R and S are arbitrary rings such that there is a surjective Morita context for these rings, then the categories R-Mod and S-Mod are equivalent (and the rings R and S are said to be Morita-equivalent). It is proved in Proposition 2.3 of García and Simón (Citation1991) that the converse implication holds for idempotent rings.

Recall that an idempotent ring is a ring R such that R ² = R. For an idempotent ring R we denote by R-Mod the full subcategory of the category of all left R-modules whose objects are the “unital” nondegenerate modules. Here a left R-module is said to be unital if M = RM, and is said to be nondegenerate if, for m ∈ M, Rm = 0 implies m = 0. Note that if R has an identity, then R-Mod is the usual category of left R-modules.

The following result can be found in García and Simón (Citation1991) (see Proposition 2.5 and Theorem 2.7).

4.1. Theorem

Let R and S be two idempotent rings. Then R-Mod and S-Mod are equivalent categories if and only if there exists a Morita context (R, S, M, N, ϕ, ψ), with M ∈ R-Mod ∩ Mod-S, N ∈ S-Mod ∩ Mod-R, and ϕ and ψ surjective.

4.2. Remark

If (R, S, M, N, ϕ, ψ) is a Morita context for two idempotent rings R and S, with M ∈ R-Mod ∩ Mod-S and N ∈ S-Mod ∩ Mod-R, and T is the Morita ring of the context, then (M, N) is an associative pair and if R has no total left or right zero divisors and S has no total left or right zero divisors, then T is the envelope of the associative pair.

4.3. Theorem

Let R and S be two Morita-equivalent idempotent rings such that R has no total left or right zero divisors and S has no total right zero divisors, and let T = (R, S, M, N) be the Morita ring of the context. Then the following are equivalent conditions:

Proof

Notice that by 4.2, T is the envelope of the associative pair A. Since the modules are left and right nondegenerate, and the rings are idempotent, we have that A has neither total left right zero divisors nor total right zero divisors. Apply (1.10) to obtain (i) ⇔ (ii) ⇔ (iii).

By 1.9, (iii) ⇔ (iv).

The conditions over R and S in the previous theorem cannot be dropped, since there exist two Morita equivalent idempotent rings R and S such that R is left nonsingular while Z _l (S) ≠ 0 (consider R and S = R/J, with R and J as in the following lemma).

4.4. Lemma

Let R be a commutative idempotent ring, and consider an ideal J of R such that JR = 0 and R/J is semiprime. Then , with product given by

Proof

It is not difficult to prove that the product is well defined. By the idempotency of R, (R/J)² = R/J. Moreover, given r ∈ R = R ², , with α ∈ ℕ and m _i , y _i  ∈ R. Hence, . This proves the surjectivity. The modules of the context are unital by the idempotency of R. Finally, we will prove that the modules are nondegenerate. Indeed for every r ∈ R implies and by the semiprimeness of . The semiprimeness of R¯ implies too that R/J is a nondegenerate R/J-module.

We recall that a ring R is said to have finite left local Goldie dimension if any element of R has finite left Goldie (or uniform) dimension. The left Goldie dimension of an element a ∈ R will be denoted by u-dim(a).

4.5. Theorem

Let R and S be two Morita-equivalent idempotent rings such that R has no total left or right zero divisors and S has no total left or right zero divisors, and suppose R left nonsingular (equivalently S left nonsingular). Let T = (R, S, M, N) be the Morita ring of the context, and define A ≔ (M, N). Then the following are equivalent conditions:

Proof

Fix the following notation: (ℰ, e) is the standard imbedding of the associative pair A, , which exists by (1.8) and (2.9), 𝒜₁₁ = R, 𝒜₂₂ = S and 𝒜 = T.

(1) 𝒬 _ii is a left quotient ring of 𝒜 _ii .

Take, for example, p ₁₁, q ₁₁ ∈ 𝒬₁₁, with p ₁₁ ≠ 0. Since 𝒬 is a left quotient ring of 𝒜, we can choose a ∈ 𝒜 such that a p ₁₁ ≠ 0 and a q ₁₁ ∈ 𝒜. If a ₁₁ p ₁₁ ≠ 0, then we have finished. Suppose a ₂₁ p ₁₁ ≠ 0. The absence of total right zero divisors in A implies b ₁₂ a ₂₁ p ₁₁ ≠ 0 for some b ₁₂ ∈ 𝒜₁₂. Then the element c ₁₁ ≔ b ₁₂ a ₂₁ ∈ 𝒜₁₁ satisfies: c ₁₁ p ₁₁ ≠ 0 and c ₁₁ q ₁₁ ∈ 𝒜.

(2) I(𝒜 _ii ) = I(𝒜) ∩ 𝒜 _ii  = I(ℰ) ∩ 𝒜 _ii .

For an element a ₁₁ ∈ 𝒜₁₁ we have:

(v) ⇒ (i), (ii), (iii), (iv) follows by 3.3(ii) and by (2).

(i), (ii), (iii) or (iv) ⇒ (v) by 3.3(ii) and (2), I(𝒜 _ij ) = I(𝒜) ∩ 𝒜 _ij , for i, j = 1, 2, i ≠ j. Taking into account that I(𝒜) is an ideal of 𝒜 (by 3.2) and that 𝒜 is generated, as an ideal of ℰ, by 𝒜 _ij , the result follows.

Let 𝒜₁₁ and 𝒜₂₂ be two Morita-equivalent idempotent rings, denote the Morita ring of the context by 𝒜 = (𝒜₁₁, 𝒜₂₂, 𝒜₁₂, 𝒜₂₁), and suppose that there exists and (as under the hypothesis of 4.3 and 4.5). The natural questions that arise are the following: are these two rings Morita-equivalent too?, and, if , do 𝒬₁₁ and 𝒬₂₂ coincide with and , respectively? The answer is negative in both cases.

4.6. Example

Consider a simple and non unital ring ℛ which coincides with its socle, and take a minimal idempotent e ∈ ℛ. Then is a Morita context for the idempotent rings eℛe and ℛ which have no total right zero divisors. On the one hand, by Proposition 4.3.7 of Lambek (Citation1976), , with V a left vector space over a division ring Δ (which is isomorphic to eℛe). On the other hand, (because eℛe is a division ring). But En d _Δ(V) and Δ are not Morita equivalent rings because A is left nonsingular and has finite left local Goldie dimension, while En d _Δ(V), which is left nonsingular, has not finite left local Goldie dimension, and this property is Morita invariant, by virtue of 4.5.

Finally we prove that for semiprime left local Goldie rings, the Fountain–Gould left orders of two idempotent Morita equivalent rings are Morita equivalent too. This contrasts with the previous example, which shows that under the same conditions (semiprime and left local Goldie), the maximal left quotient rings of two Morita equivalent rings are not Morita equivalent.

4.7. Theorem

Let R and S be two Morita equivalent semiprime idempotent rings, with R left local Goldie. Then:

Proof

(i) follows by (4.3) and (4.5).

(ii) Consider a surjective Morita context (R, S, M, N) for the rings R and S, and let be the Morita ring of the context. Denote by Q ₁ and Q ₂ the maximal left quotient rings of R and S, respectively.

Consider the unital ring , where R ¹ and S ¹ denote the unitizations of R and S, respectively. This ring has two orthogonal idempotents and such that e + f = 1 _B and Ae + eA ⊆ A. By (2.7) of Aranda Pino et al. (CitationTo appear), there exist two orthogonal idempotents such that u + v = 1 _Q and R = uAu, S = vAv, M = uAv, and N = vAu are contained in Q. Moreover, (by Lemma 1.8 of Aranda Pino et al. (CitationTo appear), which can be used because Au + uA ⊆ A and la n _A (Au) = ra n _A (uA) = 0) . And analogously . This means that M, N, Q ₁ and Q ₂ can be considered inside Q as uQv, vQu, uQu and vQv, respectively. By 4.9 of Gómez Lozano and Siles Molina (Citation2002), T ₁ = R Q ₁ and T ₂ = S Q ₂. We claim that is a surjective Morita context for the idempotent rings R Q ₁ and S Q ₂.

R Q ₁ R Q ₁ = R Q ₁ since every element q ∈ T = R Q ₁ can be written as q = a ^#2 ab, with a, b ∈ R. The same argument for S Q ₂ shows that it is an idempotent ring. Moreover, this implies R Q ₁ R Q ₁ M Q ₂ = R Q ₁ M Q ₂ and S Q ₂ S Q ₂ N Q ₁ = S Q ₂ N Q ₁.

Now, R Q ₁ M Q ₂ = R Q ₁ M S ² Q ₂ ⊆ R Q ₁ M Q ₂ S Q ₂ ⊆ R Q ₁ M Q ₂. Hence R Q ₁ M Q ₂ S Q ₂ = R Q ₁ M Q ₂ and analogously S Q ₂ N Q ₁ R Q ₁ = S Q ₂ N Q ₁, which shows that the modules are unital.

In what follows, we will show the surjectivity of the Morita context.

R Q ₁ M Q ₂ S Q ₂ N Q ₁ ⊆ R Q ₁ = R Q ₁ R Q ₁ R Q ₁ R Q ₁ = R Q ₁ MN Q ₁ MNMN Q ₁ MN Q ₁ ⊆ R Q ₁ M Q ₂ S Q ₂ N Q ₁. Hence R Q ₁ M Q ₂ S Q ₂ N Q ₁ = R Q ₁. Analogously, S Q ₂ N Q ₁ R Q ₁ M Q ₂ = S Q ₂.

Finally, we have that the modules are nondegenerate:

Indeed, suppose . (M, N) ⊆ (R Q ₁ M Q ₂, S Q ₂ N Q ₁) by (2.9). This implies that (R Q ₁ M Q ₂, S Q ₂ N Q ₁) is a left quotient pair of (M, N). Hence, if t ≠ 0, for some (m, n) ∈ (M, N), 0 ≠ mnt ∈ M. Since M is a nondegenerate right S-module and S is idempotent, 0 ≠ mnt S ² ⊆ MNtS Q ₂. This implies R Q ₁ M Q ₂ nondegenerate as a right S Q ₂ -module and as a left R Q ₁ -module.

Now, changing the roles of R and S, the proof is complete.