Centralizing mappings, Morita context and generalized (α, β)-derivations

Peer review under responsibility of Taibah University.

Abstract

In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

MSC:

• 16W25
• 16N60
• 16U80
• 16D90

Keywords:

• Prime rings
• (α, β)-Derivations and generalized (α, β)-derivations
• Lie ideals
• Morita context

1 Introduction

In the mid-forties in [1] Jacobson proved that “for every element r in a ring R, if r^n(r) = r, for some positive integer n(r), then R is commutative”. Inspired from this result, several techniques are developed to investigate conditions under which a ring becomes commutative, for instance, generalizing Herstein’s conditions, using restrictions on polynomials, introducing derivations and generalized derivations on rings, looking at special properties for rings, etc. For more details and references see the review article [2]. One can also achieve this goal by comparing two rings and imposing conditions on them. Let us assume that rings R and S are ingredients of a Morita context. It is observed in [3] that if a Morita context is semi-projective, in the sense that the Morita map on S is epic, and if R is commutative and S is reduced, then S becomes commutative. In the present note we weakened the condition on the ring R and assume that R is prime and satisfies certain conditions on generalized (α, β)-derivations. These conditions are listed in the bottom of this section.

Unless otherwise stated, the termring is used here for an associative ring that may not necessarily contain the unity 1. We assume throughout that the datum K(R, S) = {R, S, M, N, μ_R, τ_S, I, J} is a Morita context (in short MC) in which R and S are rings, M and N are (S, R) and (R, S)-bimodules, respectively, and μ_R : N ⊗ _SM → R and τ_S : M ⊗ _RN → S are bimodule homomorphisms with the associativity conditions$m_1{\mu }_R(n\otimes m)={\tau }_S(m_1\otimes n)m,\forall m,m_1\in M,n\in N$and${\mu }_R(n\otimes m)n_1=n{\tau }_S(m\otimes n_1),\forall m\in M,n,n_1\in N$where μ_R and τ_S are called Morita maps (or MC maps). The images μ_R(N ⊗ _SM) = I and τ_S(M ⊗ _RN) = J are two-sided ideals of R and S, respectively, and are called the trace ideals of the MC. If both MC maps are epimorphisms, i.e., I = R and J = S, then K(R, S) is said to be a projective Morita context (or PMC). If one of the MC maps, either μ_R or τ_S, is an epimorphism, then K(R, S) is said to be a semi-projective Morita context or a semi-PMC [4].

Recall that a ring R is prime if for any a, b ∈ R, aRb = {0} implies that a = 0 or b = 0 and semiprime if for any a ∈ R, aRa = {0}, which implies that a = 0 (or equivalently, R has no non-zero nilpotent ideals). A ring R is called reduced if R has no non-zero nilpotent elements. Clearly, every reduced ring is semiprime and every prime ring is semiprime, but the converse of these statements need not be true in general.

For each x, y ∈ R, denote the commutator xy − yx by [x, y] and the anti-commutator xy + yx by x ∘ y. By a derivation on R we mean the most natural derivation d : R ⟶ R which is additive as well as satisfying the relation d(xy) = d(x)y + xd(y) for all x, y ∈ R. In particular, for a fixed a ∈ R, the mapping I_a : R ⟶ R given by I_a(x) = [x, a] is a derivation, called an inner derivation of R.

Let us review some generalizations of the notion of derivation in rings. Let α and β be endomorphisms of R. An additive map d : R ⟶ R is called an (α, β)-derivation if d(xy) = d(x)α(y) + β(x)d(y) holds for all x, y ∈ R. Note that a (1, 1)-derivation is a derivation, where 1 is the identity map on R. We denote xα(y) − β(x)y by [x, y]_α,β and xα(y) + β(x)y by (x ∘ y)_α,β for all x, y ∈ R.

An example of an (α, β)-derivation: when R has a unity 1 and a nontrivial central idempotent e, set d(x) = ex, α(x) = (1 − e)x and β = 1 (or d). Here d is an (α, β)-derivation, but not a derivation, because d(ee) = eee ≠ 2eee = (ee)e + e(ee) = d(e)e + ed(e). In any ring with endomorphism β, if we set d = 1 − β, then d is a (β, 1)-derivation, but not a derivation when R is semiprime, unless β = 1.

For a fixed a, the map d_a : R ⟶ R given by d_a(x) = [a, x]_α,β for all x ∈ R is an (α, β)-derivation which is said to be an (α, β)-inner derivation. An additive mapping F : R ⟶ R is called a generalized (α, β)-inner derivation if F(x) = aα(x) + β(x)b, for some fixed a, b ∈ R and for all x ∈ R. A simple computation yields that if F is a generalized (α, β)-inner derivation, then for all x, y ∈ R, we have

$F(\mathrm{xy})=F(x)\alpha (y)+\beta (x)d_{-b}(y),$where d_−b is an (α, β)-inner derivation. With this viewpoint, an additive map F : R ⟶ R is called a generalized (α, β)-derivation associated with an (α, β)-derivation d : R ⟶ R such that$F(\mathrm{xy})=F(x)\alpha (y)+\beta (x)d(y)\text{holds for all}x,y\in R.$Clearly, this notion includes those of (α, β)-derivation when F = d, and of generalized derivation, when the case.

An additive subgroup L of R is said to be a Lie ideal of R if [L, R] ⊆ L. Clearly, every ideal is a Lie ideal but the converse need not be true in general. In Section 2 we have established some lemmas by involving Lie ideals and have used them to obtain the main results in Section 3.

In Section 3, in stead of using commutativity of the ring R, as in [3], here we have weakened it by assuming that R is prime and satisfies any one of the following conditions on generalized (α, β)-derivations: (i) [F(x), x]_α,β − [y, x]_α,β ∈ Z(R), (ii) (F(x) ∘ x)_α,β − (y ∘ x)_α,β ∈ Z(R), (iii) (F(x) ∘ F(y)) ∈ Z(R), (iv) [F(x), F(y)] − [G(x), y]_α,β ∈ Z(R), and (v) F(x) ∘ F(y) − (G(x) ∘ y)_α,β ∈ Z(R) for all x, y ∈ I.

2 Preliminary results

For the sake of interest, some results in this section are stated in a more general setting, that is, in terms of Lie ideals. Their application is restricted to ideals in the last section.

Following are some useful identities which hold for every x, y, z ∈ R. We will use them in the proof of our theorems.

•	[xy, z]α,β = x[y, z]α,β + [x, β(z)]y = x[y, α(z)] + [x, z]α,βy;

•	[x, yz]α,β = β(y)[x, z]α,β + [x, y]α,βα(z);

•	(x ∘ (yz))α,β = (x ∘ y)α,βα(z) − β(y)[x, z]α,β = β(y)(x ∘ z)α,β + [x, y]α,βα(z);

•	((xy) ∘ z)α,β = x(y ∘ z)α,β − [x, β(z)]y = (x ∘ z)α,βy + x[y, α(z)].

The proof of Remark 2.1 can be verified easily.

Remark 2.1

Let R be a prime ring. For a nonzero element a ∈ Z(R), if ab ∈ Z(R), then b ∈ Z(R).

We begin our discussion with the following results:

Lemma 2.1

[5], Lemma 4

If R is a prime ring and I be a nonzero ideal of R such that a, b ∈ R such that aIb = {0}, then a = 0 or b = 0.

Lemma 2.2

[6], Lemma 2.4

Let R be a prime ring, and L be a nonzero Lie ideal of R. Let α, β be automorphisms of R. If [x, y]_α,β = 0 for all x, y ∈ L, then L ⊆ Z(R).

Lemma 2.3

[6], Lemma 2.5

Let R be a prime ring and L be a nonzero Lie ideal of R. Let α, β be automorphisms of R such that α(L) ⊆ L. If [x, y]_α,β ∈ Z(R), for all x, y ∈ L then L is a subset of the center of R.

Lemma 2.4

[6], Lemma 2.6

Let R be a prime ring and L be a nonzero Lie ideal of R. Let α, β be automorphisms of R such that β(L) ⊆ L. If (x ∘ y)_α,β ∈ Z(R), then L ⊆ Z(R).

Lemma 2.5

Let R be a 2-torsion free semiprime ring and L be a nonzero Lie ideal of R. Then Z(L) ⊆ Z(R).

Proof

Suppose that c ∈ Z(L), then $[c,v]=0$ for all $v\in L$. Replacing $v$ by [c, r] for all r ∈ R, we get [c, [c, r]] = 0, again replacing r by rs, we get [c, [c, r]s + r[c, s]] = 0 for all s ∈ R and hence 2[c, r][c, s] = 0 since R is a 2-torsion free, we get [c, r][c, s] = 0. Now, replacing s by sr, we get [c, r]R[c, r] = 0 and hence c ∈ Z(R). □

In view of the above lemma, we immediately get the following lemma:

Lemma 2.6

Let R be a 2-torsion free semiprime ring and L be a Lie ideal of R. If α : R ⟶ R be a ring of epimorphism on R such that 0 ≠ c ∈ Z(L), then α(c) ∈ Z(R).

3 Centralizing generalized (α, β)-derivations via Morita contexts

In this section we will prove the main results and will discuss some consequences. Throughout we assume that {0} ≠ d(Z(R)) ⊆ Z(R), {0} ≠ g(Z(R)) ⊆ Z(R) and K(R, S) = {R, S, M, N, μ_R, τ_S, I, J} is an MC in which I and J are trace ideals. Because trace ideals are two-sided ideals, these are Lie ideals. Hence, in the following, we can freely apply above lemmas in which Lie ideals are involved.

We begin with the following lemma which will be used to prove several results.

Lemma 3.1

[3], Corollary 2.3

Let R and S be rings of an MC K(R, S) and let I and J be the trace ideals of R and S respectively. If I ⊆ Z(R) and S is reduced, then J ⊆ Z(S).

Theorem 3.1

Let K(R, S) be an MC in which the trace ideals I and J are non-zero. Suppose that α, β are automorphisms of R such that α(I), β(I) ⊆ I, and R admits a generalized (α, β)-derivation F with associated (α, β)-derivation d such that either

(i)	[F(x), x]α,β − [y, x]α,β ∈ Z(R) for all x ∈ I, or

(ii)	(F(x) ∘ x)α,β − (y ∘ x)α,β ∈ Z(R) for all x ∈ I, or

(iii)

F(x) ∘ F(y) ∈ Z(R) for all x, y ∈ I.

Then

(a)	If R is prime and S is reduced, then J is a subset of the center of S.

(b)	If R is prime, S is reduced, and if τS is epic, then S becomes commutative.

Proof

(a) (i) For all x, y ∈ I, we have

(3.1) ${[F(x),x]}_{\alpha ,\beta }-{[y,x]}_{\alpha ,\beta }\in Z(R).$(3.1) If F = 0, then [y, x]_α,β ∈ Z(R) for all x, y ∈ I, thus I ⊆ Z(R) by Lemma 2.3. Since R is prime and S is reduced, we find that J ⊆ Z(S) by Lemma 3.1.

Therefore, we shall assume that d ≠ 0. Linearizing (3.1)(3.1) ${[F(x),x]}_{\alpha ,\beta }-{[y,x]}_{\alpha ,\beta }\in Z(R).$(3.1) , we find that

(3.2) ${[F(x),z]}_{\alpha ,\beta }+{[F(z),x]}_{\alpha ,\beta }\in Z(R).$(3.2) For any nonzero c ∈ Z(R), replacing z by zc in (3.2)(3.2) ${[F(x),z]}_{\alpha ,\beta }+{[F(z),x]}_{\alpha ,\beta }\in Z(R).$(3.2) and using (3.2)(3.2) ${[F(x),z]}_{\alpha ,\beta }+{[F(z),x]}_{\alpha ,\beta }\in Z(R).$(3.2) and Lemma 2.6, we get$\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){[d(c),x]}_{\alpha ,\beta }+\beta ([z,x])d(c)\in Z(R),$Again, replacing z by yz in the above expression, we get$\beta (y)\{\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){[d(c),x]}_{\alpha ,\beta }+\beta ([z,x])d(c)\}+\beta ([y,x]z)d(c)\in Z(R).$Thus, in particular, we have$[\beta (y)\{\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){[d(c),x]}_{\alpha ,\beta }+\beta ([z,x])d(c)\}+\beta ([y,x]z)d(c),\beta (y)]=0$This gives

(3.3) $[\beta ([y,x]z)d(c),\beta (y)]=0\text{for all}x,y,z\in I$(3.3) This can be rewritten as β([[y, x]z, y])d(c) = 0. Since {0} ≠ d(Z(R)) ⊆ Z(R) and R is prime, we get β([[y, x]z, y]) = 0 for all x, y, z ∈ I, that is [y, x][z, y] + [[y, x], y]z = 0. Again replacing z by zx and using the above expression, we get [y, x]z[x, y] = 0. Thus, by Lemma 2.1 forces that [x, y] = 0 for all x, y ∈ I and hence I ⊆ Z(R) by Lemma 2.2. Since R is prime and S is reduced, we get J ⊆ Z(S) by Lemma 3.1.

(ii) It is given that F is generalized (α, β)-derivation on R. If F = 0, then (y ∘ x)_α,β ∈ Z(R), then by Lemma 2.4 I ⊆ Z(R). Since S is reduced, J ⊆ Z(S) by Lemma 3.1.

Henceforth, we shall assume that d ≠ 0. Now for all x, y ∈ I, we have${(F(x)\circ x)}_{\alpha ,\beta }-{(y\circ x)}_{\alpha ,\beta }\in Z(R).$Linearizing the above expression, we find that

(3.4) ${(F(x)\circ z)}_{\alpha ,\beta }+{(F(z)\circ x)}_{\alpha ,\beta }\in Z(R)\text{for all}x,z\in I.$(3.4) For any nonzero c ∈ Z(R), replacing z by zc in (3.4)(3.4) ${(F(x)\circ z)}_{\alpha ,\beta }+{(F(z)\circ x)}_{\alpha ,\beta }\in Z(R)\text{for all}x,z\in I.$(3.4) , we get${(F(x)\circ z)}_{\alpha ,\beta }\alpha (c)-\beta (z){[F(x),c]}_{\alpha ,\beta }+{(F(z)\circ x)}_{\alpha ,\beta }\alpha (c)+F(z)[\alpha (c),\alpha (x)]+\beta (z){(d(c)\circ x)}_{\alpha ,\beta }-[\beta (z),\beta (x)]d(c)\in Z(R).$Now, applications of (3.4)(3.4) ${(F(x)\circ z)}_{\alpha ,\beta }+{(F(z)\circ x)}_{\alpha ,\beta }\in Z(R)\text{for all}x,z\in I.$(3.4) and Lemma 2.6, yields that$-\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){(d(c)\circ x)}_{\alpha ,\beta }-\beta ([z,x])d(c)\in Z(R).$Again replacing z by yz in the last equation, we get$\beta (y)\{-\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){(d(c)\circ x)}_{\alpha ,\beta }-\beta [z,x]d(c)\}-\beta ([y,x]z)d(c)\in Z(R).$Thus, in particular, we have$[\beta (y)\{-\beta (z){[F(x),c]}_{\alpha ,\beta }+\beta (z){(d(c)\circ x)}_{\alpha ,\beta }-\beta [z,x]d(c)\}-\beta ([y,x]z)d(c),\beta (y)]=0.$This gives [β([y, x]z)d(c), β(y)] = 0 for all x, y, z ∈ I. Now, applying similar technique as used after (3.3)(3.3) $[\beta ([y,x]z)d(c),\beta (y)]=0\text{for all}x,y,z\in I$(3.3) in the proof of (i), yields the required result.

(iii) For all x, y ∈ L we have

(3.5) $F(x)\circ F(y)\in Z(R).$(3.5) For any nonzero c ∈ Z(R) replacing y by yc in (3.5)(3.5) $F(x)\circ F(y)\in Z(R).$(3.5) and using (3.5)(3.5) $F(x)\circ F(y)\in Z(R).$(3.5) and Lemma 2.6, we get (F(x) ∘ β(y))d(c) ∈ Z(R). Since {0} ≠ d(Z(R) ⊆ Z(R) and R is prime, we get F(x) ∘ β(y) ∈ Z(R). Again, for any nonzero c ∈ Z(R), replacing x by xc in the above expression and using Lemma 2.6, we get β(x ∘ y)d(c) ∈ Z(R) for all x, y ∈ I. Again, Since R is prime and {0} ≠ d(Z(R) ⊆ Z(R), we obtain x ∘ y ∈ Z(R) for all x, y ∈ I, that is, (x ∘ y) ∈ Z(R) for all x, y ∈ I. This implies that [(x ∘ y), r] = 0 for all r ∈ R. Replacing y by yx, we find (x ∘ y)[x, r] = 0. For any s ∈ R, replace r by sr, to get (x ∘ y)R[x, r] = {0}. Thus, for each x ∈ I either (x ∘ y) = 0 or [x, r] = 0. Let A = {x ∈ I ∣ (x ∘ y) = 0 for all y ∈ I} and B = {x ∈ I ∣ [x, r] = 0 for all r ∈ R}. Then A and B are additive subgroups of L whose union is L. But a group cannot be the union of its two proper subgroups and hence either (x ∘ y) = 0 for all x, y ∈ I or [x, r] = 0 for all x ∈ I and r ∈ R. If (x ∘ y) = 0, then replace x by [x, r]x, to get [x, r][x, y] = 0 for all x, y ∈ I and r ∈ R, again replacing r by rs, we get [x, s]R[x, y] = 0 for all x, y ∈ I, s ∈ R and primeness of R forces that either [x, s] = 0 or [x, y] = 0. If [x, s] = 0 for all x ∈ I, s ∈ R then I ⊆ Z(R). On the other hand, if [x, y] = 0 for all x, y ∈ I then by Lemma 2.2, we obtain that I ⊆ Z(R). Since S is reduced, J ⊆ Z(S) by Lemma 3.1.

(b) In either case of (i), (ii) or (iii), if τ_s is epic, then S = J ⊆ Z(S). Hence S is commutative.

Theorem 3.2

Let K(R, S) be an MC in which the trace ideals I and J are non-zero. Suppose that α, β are automorphisms of R such that α(I), β(I) ⊆ I, and R admits generalized (α, β)-derivations F and G with associated (α, β)-derivations d and g respectively, such that F = 0 (or G = 0) or d ≠ 0 (or g ≠ 0) and R satisfies any one of the following properties

(i)	[F(x), F(y)] − [G(x), y]α,β ∈ Z(R) for all x, y ∈ I,

(ii)	F(x) ∘ F(y) − (G(x) ∘ y)α,β ∈ Z(R) for all x, y ∈ I.

Then

(a)	If R is prime and S is reduced, then J is a subset of the center of S.

(b)	If R is prime, S is reduced, and if τS is epic, then S becomes commutative.

Proof

(a) (i) For any x, y ∈ I, we have

(3.6) $[F(x),F(y)]-{[G(x),y]}_{\alpha ,\beta }\in Z(R).$(3.6) If G = 0, then we have [F(x), F(y)] ∈ Z(R) for all x, y ∈ I. Thus, by Theorem 3.3(i) of [6], it yields the required result.

If F = 0, then we have [G(x), y]_α,β ∈ Z(R) for all x, y ∈ I. Thus, in particular [G(x), x]_α,β ∈ Z(R) for all x ∈ I and hence by Theorem 3.1(i) of [6], we get the required result.

Henceforth, we shall assume that d ≠ 0 and g ≠ 0. Thus, for any nonzero c ∈ Z(R) replacing y by yc in (3.6)(3.6) $[F(x),F(y)]-{[G(x),y]}_{\alpha ,\beta }\in Z(R).$(3.6) and using (3.6)(3.6) $[F(x),F(y)]-{[G(x),y]}_{\alpha ,\beta }\in Z(R).$(3.6) and Lemma 2.6, we get$[F(x),\beta (y)]d(c)-\beta (y){[G(x),c]}_{\alpha ,\beta }\in Z(R).$Again replacing y by wy in the last expression, we get$\beta (w)\{[F(x),\beta (y)]d(c)-\beta (y){[G(x),c]}_{\alpha ,\beta }\}+[F(x),\beta (w)]\beta (y)d(c)\in Z(R).$Thus, in particular, we have$[\beta (w)\{[F(x),\beta (y)]d(c)-\beta (y){[G(x),c]}_{\alpha ,\beta }\}+[F(x),\beta (w)]\beta (y)d(c),\beta (w)]=0.$This gives

(3.7) $[[F(x),\beta (w)]\beta (y)d(c),\beta (w)]=0\text{for all}x,y,w\in I.$(3.7) This can be rewritten as $[[F(x),\beta (w)]\beta (y),\beta (w)]d(c)=0$. Since o {0} ≠ d(Z(R)) ⊆ Z(R) and R is prime, we find that $[[F(x),\beta (w)]\beta (y),\beta (w)]=0$. That is$[F(x),\beta (w)][\beta (y),\beta (w)]+[[F(x),\beta (w)],\beta (w)]\beta (y)=0.$Again replacing y by ym in the last expression, we get$[F(x),\beta (w)]\beta (y)\beta [m,w]=0\text{for all}x,y,w,m\in I,$and hence ${\beta }^{-1}([F(x),\beta (w)])I[m,w]=\{0\}$ for all $x,m,w\in I$, and applying Lemma 2.1 and the fact that (I, +) is not the union of its two proper subgroups show that either ${\beta }^{-1}([F(x),\beta (w)])=0$ or $[m,w]=0$ for all $x,m,w\in I$. If $[m,w]=0$ for all $m,w\in I$, then I ⊆ Z(R) by Lemma 2.2. On the other hand, if $[F(x),\beta (w)]=0$ for all $x,w\in I$, for any nonzero c ∈ Z(R), replacing x by xc and by Lemma 2.6, we get $\beta ([x,w])d(c)=0$ for all $x,w\in I$. Now, primeness of R and {0} ≠ d(Z(R)) ⊆ Z(R) forces that $\beta ([x,w])=0$ and hence again by Lemma 2.2, I ⊆ Z(R). Hence in both the cases, we get I ⊆ Z(R). Since S is reduced, J ⊆ Z(S) by Lemma 3.1.

(ii) It is given that F and G are generalized (α, β)-derivations on R associated with (α, β)-derivations d and g respectively, such that F(x) ∘ F(y) − (G(x) ∘ y)_α,β ∈ Z(R) for all x, y ∈ I. If G = 0 then we have F(x) ∘ F(y) ∈ Z(R) for all x, y ∈ I. Now by Theorem 3.1(ii), we get that I ⊆ Z(R). Since S is reduced, J ⊆ Z(S) by Lemma 3.1.

If F = 0 then we have (G(x) ∘ y)_α,β ∈ Z(R) for all x, y ∈ I. Thus, in particular (G(x) ∘ x)_α,β ∈ Z(R) for all x ∈ I and hence by Theorem 3.1(iii) of [6], we get the required result.

Therefore, we shall assume that d ≠ 0 and g ≠ 0. For any x, y ∈ I we have

(3.8) $F(x)\circ F(y)-{(G(x)\circ y)}_{\alpha ,\beta }\in Z(R).$(3.8) For any nonzero c ∈ Z(R), replacing y by yc in (3.8)(3.8) $F(x)\circ F(y)-{(G(x)\circ y)}_{\alpha ,\beta }\in Z(R).$(3.8) and using Lemma 2.6, we get$(F(x)\circ \beta (y))d(c)+\beta (y){[G(x),c]}_{\alpha ,\beta }\in Z(R)\text{for all}x,y\in I.$Again, replacing y by wy in the last expression, we get$\beta (w)\{(F(x)\circ \beta (y))d(c)+\beta (y){[G(x),c]}_{\alpha ,\beta }\}+[F(x),\beta (w)]\beta (y)d(c)\in Z(R).$In particular, we have$[\beta (w)\{(F(x)\circ \beta (y))d(c)+\beta (y){[G(x),c]}_{\alpha ,\beta }\}+[F(x),\beta (w)]\beta (y)d(c),\beta (w)]=0.$This gives$[[F(x),\beta (w)]\beta (y)d(c),\beta (w)]=0\text{for all}x,y,w\in I.$Note that the last relation is same as the relation (3.7)(3.7) $[[F(x),\beta (w)]\beta (y)d(c),\beta (w)]=0\text{for all}x,y,w\in I.$(3.7) . Hence, by similar arguments we get the required result.

(b) By the same argument as above, in the cases either (i) or (ii), if τ_s is epic, then S = J ⊆ Z(S). Hence S is commutative.

Remark 3.1

Let K(R, S) be a PMC in which rings R and S are equipped with multiplicative identity 1. Then Z(R) ≅ Z(S). If the conditions of either Theorem 3.1 or 3.2, are satisfied, then S ≅ Z(R). Hence R can be treated as an S − Algebra. Moreover in this case S becomes prime, as being prime is a Morita invariant property.

Remark 3.2

Let K(R, S) be a semi-PMC in which τ_S is epic. Then the generalized matrix ring $T=\left[\begin{array}{cc}\hfill R\hfill & \hfill M\hfill \\ \hfill N\hfill & \hfill S\hfill \end{array}\right]$ and S are Morita equivalent [4, Theorem 2.1]. Hence, trivially, in this case, if the conditions of either of Theorems 3.1 or 3.2 are satisfied, then Z(T) ≅ S.

Notes

Peer review under responsibility of Taibah University.