Morita context and generalized ( α , β ) − derivations

Let R and S be rings of a semi-projective Morita context, and α, β be automorphisms of R. An additive mapping F : R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)-derivation d: R → R such that F (xy) = F (x)α(y)+β(x)d(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y]α,β = xα(y) − β(y)x and (x ◦ y)α,β = xα(y) + β(y)x. In the present paper, we shall show that if the ring S is reduced then it is a commutative, in a compatible way with the ring R. Also, we obtain some results on bi-algebras via Cauchy modules.

A classical problem in ring theory is to study and generalized conditions under which a ring becomes commutative.So far the best tools found for this purpose are the derivations on rings and also on their modules.We can also achieve this goal by comparing two rings and impose conditions on them.If one of the rings is commutative, in compatible way, the other ring will also become commutative.In order to explore these ideas Morita theory is found to be a suitable tool.
Unless otherwise stated the term rings with center Z(R) is used here for associative rings.We assume throughout that the datum K(R, S) = {R, S, M, N, µ R , τ S } is said to be Morita context (or MC) in which R and S are rings, M and N are (S, R) and (R, S) bimodules, respectively, µ R : N ⊗ S M → R and τ S : M ⊗ R N → S are bimodule homomorphisms with associative condition where µ R and τ S are called Morita maps (or MC maps).The images µ R := I and τ R := J are two-sided ideals of R and S, respectively, and are called the trace ideals of the MC.If both MC maps are epimorphism, 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 is an epimorphism, then K(R, S) is said to be semi-projective Morita context or semi-PMC.If K(R, S) is PMC rings, then the rings R and S are said to be Morita equivalent.
An algebra over R (or an R−algebra) is an (R, R)−bimodule M together with module morphisms (we will also call them linear maps): Let R be a commutative ring.An R−coalgebra is an (R, R)−bimodule C, with R−linear maps: Now we combine the notions of R−algebras and R−coalgebras.An R−bialgebra B is an R−module together with algebra and coalgebra structures.
Let R and S be rings and M an (R, S)−bimodule.Then the dual of M which is denoted by M * = Hom R (M, R) is an (S, R)−bimodule, and for every left Rmodule L there is a canonical module morphism defined by Morita context and generalized (α, β)−derivations

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If ϕ M L is an isomorphism for each left R−module L, then R M S is called a Cauchy module.(see [1] and [5]).
For each x, y ∈ R, denote the commutator xy − yx by [x, y] and the anticommutator xy + yx by x • y.Recall that a ring R is prime if for any a, b ∈ R, aRb = {0} implies that a = 0 or b = 0.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.
To understand our results it is best to review some generalizations of the notion of derivation in rings.Let α and β be the 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. A (1, 1)-derivation is called simply a derivation, where 1 is the identity map on R.An example of an (α, β)-derivation, when R has a nontrivial central idempotent e is to let d(x) = ex, α(x) = (1 − e)x and β = 1 (or d).Here, d is 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 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 (xy) = F (x)α(y) + β(x)d(y) holds for all x, y ∈ R.
Clearly this notion includes those of (α, β)-derivation when F = d, of derivation when F = d and α = β = 1, and of generalized derivation, when is the case Clearly, every ideal is a Lie ideal but the converse need not be true in general.A Lie ideal L is said to be square closed if a 2 ∈ L for all a ∈ L. In Section 2 we have established one lemma by involving square closed Lie ideals and have used them to obtain the main results in Section 3.
In the present paper our aim is to prove that if R and S are rings of a semi-PMC, in which R admits generalized (α, β)−derivations F and G satisfying certain differential identities in rings such that S is reduced, then S is commutative.Some results related to division rings, bi-algebras and Cauchy module are stated and proved.

Preliminaries Results
Following are some useful identities which hold for every x, y, z ∈ R. We will use them in the proof of our theorems.
Let us consider three important remarks.
Remark 2.1.Let R be a prime ring and H an additive subgroups of R. Let f : H → R and g : H → R be additive functions such that f (s)Rg(s) = {0} for all s ∈ H. Then either f (s) = 0 for all s ∈ H, or g(s) = 0 for all s ∈ H.

Remark 2.2. Let R and S be rings of an MC
Remark 2.3.Let R be any ring and I be a nonzero ideal of R.
The proof of Remark 2.1 is rather elementary and is based on the fact that a group cannot be written as the set-theoretic union of its two proper subgroups.Also the proof of Remark 2.2 is clear by using elementary properties of bimodules and the definition of MC.Similarly, Remark 2.3 can also be verified easily.
We begin our discussion with the following results.For the sake of interest, Lemma 2.5 in this section is stated in more general setting, that is, in terms of Lie ideals.Their application is restricted to ideals in the last section.Corollary 2.6.Let R be a prime ring with char(R) = 2, and I be a nonzero ideal of R. Let α, β be automorphisms of R. If [x, y] α,β = 0, for all x, y ∈ I, then R is commutative.
Lemma 2.7 ( [9], Theorem 2.1).Let R and S be rings of semi-PMC K(R, S) in which τ S epic.If R is commutative and S is reduced, then S is also commutative.Proof: For all x, y ∈ I, we have Further, if R is a prime ring of characteristic different from two and S is reduced, then S is commutative.
Proof: (i) For all x ∈ I, we have (3.1) Linearizing (3.1), we get For any z ∈ Z(R), replacing y by yz in (3.2), using (3.2), and Remark 2.3, we get Again, replacing y by my with m ∈ I and using the above expression, we get Thus, in particular, we have (ii) For all x ∈ I, we have Linearizing (3.4), we get For any nonzero z ∈ Z(R), replacing y by yz in the last expression and using Remark 2.3, we get Again replacing y by my, we get Thus, in particular, we have Further, if R is a prime ring of characteristic different from two and S is reduced, then S is commutative.
Proof: (i) For all x, y ∈ I, we have If F = 0, then [x, y] α,β ∈ Z(R) for all x, y ∈ I, thus R is commutative by Lemma 2.10.Since S is reduced then by Lemma 2.7, S is commutative.
Therefore, we shall assume that d = 0.For any nonzero z ∈ Z(R), since α(z) ∈ Z(R) by Remark 2.3, replacing y by yz in (3.5) and using (3.5), we get Again, replacing y by my in (3.6), we find that Thus, in particular This implies that [β([x, m])β(y)d(z), β(m)] = 0 for all x, y, m ∈ I. Notice that the arguments given in the last paragraph of the proof of Theorem 3.1 (i) are still valid in the present situation, and hence repeating the same process, we get the required result.
(ii) It is given that F is a generalized (α, β)-generalized derivation on R. If F = 0, then (x • y) α,β ∈ Z(R), thus R is commutative by Lemma 2.11.Since S is reduced so we get the required result by Lemma 2.7.
Therefore, we shall assume that d = 0. Now for all x, y ∈ I, we have For any nonzero z ∈ Z(R), replacing y by yz in (3.7) and using (3.7), we get Again replacing y by my in (3.8), we get Thus, in particular (iii) It is given that F is a generalized (α, β)-derivation on R. If F = 0, then (x • y) α,β ∈ Z(R), thus by Lemma 2.11 R is commutative.Since S is reduced so S is commutative by Lemma 2.7.
Hence we shall assume that d = 0. Now for all x, y ∈ I, we have (3.9) Replacing y by yz for any z ∈ Z(R) in (3.9), we get , for all x, y ∈ I.
Now, applying similar technique as used after (3.6) in the proof of (i) yields the required result.
(iv) It is given that F is a generalized (α, β)-derivation.If F = 0, then [x, y] α,β ∈ Z(R), for all x, y ∈ I, and hence R is commutative by Lemma 2.10.Since S is reduced so S is commutative by Lemma 2.7.
Therefore, we shall assume that d = 0.For all x, y ∈ I, we have (3.10) Replacing y by yz for any z ∈ Z(R) in (3.10), we get The last expression is the same as the equation (3.8) and hence the result follows.
(v) It is given that F is a generalized (α, β)-derivation.If F = 0, then [x, y] α,β ∈ Z(R), for all x, y ∈ I, and hence R is commutative by Lemma 2.10.Since S is reduce so we get the required result by Lemma 2.7.
Therefore, we shall assume that d = 0. Now for all x, y ∈ I, we have Replace y by yz for any z ∈ Z(R), in (3.11), to get Again replacing y by my with m ∈ I in the last expression, we get Further, if R is a prime ring of characteristic different from two and S is reduced, then S is commutative.
, for all x, y ∈ I, and hence by Lemma 2.10 R is commutative.Since S is reduced so we get the required result by Lemma 2.7.
Now we shall assume that g = 0. Then for all x, y ∈ I, we have  Proof: Suppose that M ⊗ R N is an R−bialgebra, since R is commutative and R ∼ = S from the above theorems, then by Remark 2.2 and Lemma 2.9, respectively, the datum {R, M, N, µ R } is MC.On the other hand, if the datum K(R, S) is MC, R is commutative and R ∼ = S by the above theorems, then by Remark 2.2 the datum {R, M, N, µ R } is MC, thus by Lemma 2.9 M ⊗ R N is an R−bialgebra.2

Lemma 2 . 4 (
[12], Lemma 2.1).Let R be a prime ring and I be a nonzero ideal of R. If [x, y] ∈ Z(R) for all x, y ∈ I or if (x • y) ∈ Z(R) for all x, y ∈ I, then R is commutative.
Morita context and generalized (α, β)−derivations 157 In view of Lemma 2.5 we get the following corollary:

Lemma 2 . 8 (
[9], Corollary 2.4).Let K(R, S) be a PMC of rings in which R is commutative.Then (a) If S is a reduced ring, then R is also reduced and R ∼ = S.(b) If S is a domain, then both R and S become isomorphic integral domains.(c)If S is a division ring, then both R and S are isomorphic field.

Lemma 2 . 9 ( 2 Lemma 2 . 11 .
[1], Theorem 3.7).Let R be a commutative ring, M and N Cauchy R−modules.Then the datum {R,M, N, µ R } is MC if and only if M ⊗ R N is a R−bialgebra.Lemma 2.10.Let R be a prime ring of characteristic different from two and I be a nonzero ideal of R. Let α, β be automorphisms of R. If [x, y] α,β ∈ Z(R) for all x, y ∈ I, then R is commutative.Proof:For any x, y ∈ I, we have[x, y] α,β ∈ Z(R).(2.1)Replacing x by xα(y) in (2.1), we get [x, y] α,β α(y) ∈ Z(R), this implies that [[x, y] α,β α(y), r] = 0 for all x, y ∈ I, r ∈ R. Thus, as an application of (2.1), we find that [x, y] α,β [α(y), r] = 0. Again replacing r by rα(m) and using the last expression, we get [x, y] α,β R[α(y), α(m)] = {0}, for all x, y, m ∈ I. Thus, by Remark 2.1, either α([y, m]) = 0 for all y, m ∈ I, or [x, y] α,β = 0 for all x, y ∈ I.In the first case R is commutative by Lemma 2.4.In the second one, R is commutative by Corollary 2.6.Let R be a prime ring of characteristic different from two and I be a nonzero ideal of R. Let α, β be automorphisms of R. If (x • y) α,β ∈ Z(R) for all x, y ∈ I, then R is commutative.

Corollary 3 . 6 .Corollary 3 . 7 .
Morita context and generalized (α, β)−derivations 165 By the same argument as Corollary 3.5, in the cases, from Theorem 3.1 to Theorem 3.4, if R and S are Morita equivalent rings, then by Lemma 2.8 (a), R is also reduced and Z(R) ∼ = Z(S).Since R and S are commutative, R = Z(R) and S = Z(S) and hence R ∼ = S.If S is division ring then S is a field.Since S is commutative division ring, by Lemma 2.8 (c), R and S are becomes isomorphic fileds.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 of Theorem 3.2, or of Theorem 3.3, or ofTheorem 3.4 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.Corollary 3.8.Let K(R, S) be a semi-PMC in which τ S is epic.Then the generalized matrix ring T = R M N S and S are Morita equivalent [10, Theorem 2.1].Hence, trivially, in this case, if the conditions of either of the Theorems 3.1, 3.2, 3.3, or 3.4, are satisfied, then Z(T ) ∼ = S.