On a Characterization of Commutativity for Prime Rings via Endomorphisms

Our aim in the present paper is to introduce new classes of endomorphisms and study their connection with commutativity of prime rings with involution of the second kind. Furthermore, we provide examples to show that the various restrictions imposed in the hypotheses of our theorems are not superfluous.


Introduction
Throughout the present paper R will denote an associative ring with center Z(R). For any x, y ∈ R, the symbol [x, y] will denote the commutator xy − yx; while the symbol x • y will stand for the anticommutator xy + yx. Recall that R is prime if aRb = {0} implies a = 0 or b = 0, R is called semiprime if, for x ∈ R, xRx = {0} implies that x = 0 and R is said to be 2-torsion free if 2x = 0, x ∈ R, implies x = 0.
An additive map * : R −→ R is called an involution if * is an anti-automorphism of order 2. An element x in a ring with involution (R, * ) is said to be hermitian if x * = x and skew-hermitian if x * = −x. The sets of all hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R), respectively. The involution is said to be of the first kind if Z(R) ⊆ H(R), otherwise it is said to be of the second kind. In the later case it is straightforward to check that S(R) ∩ Z(R) = {0}. A mapping F : R −→ R is called commutativity preserving on a subset S of R if [x, y] = 0 implies that [F (x), F (y)] = 0, for all x, y ∈ S. The mapping F is called strong commutativity preserving (SCP) on S if [F (x), F (y)] = [x, y] for all x, y ∈ S.
Given the importance of some additive mappings and their relations with the global structure of the ring, a considerable amount of work has been done on this line, for example, derivations, homomorphisms, endomorphisms and related maps during the last decades ( see for example [4] and [6] [2]) investigated commutativity of prime and semiprime rings admitting derivations and endomorphisms, which are SCP on its certain subset.
Recently, the authors in [7] introduced new classes of endomorphisms and studied their connection with commutativity of prime rings with involution of the second kind. Moreover, they provide a complete description and classification for some of these endomorphisms. In fact, they introduced the concepts of * -SCP and * -Skew SCP mappings as follows: Let R be a ring and let g : R −→ R be an endomorphism. Then g is called strong anti-commutativity preserving (SACP) if g(x) • g(y) = x • y for all x, y ∈ R. If R is equipped with an involution * , then g is called * -SCP (resp. * - Moreover, if g satisfies g(x) • g(x * ) = x • x * for all x ∈ R, then g is said to be * -SACP). Motivated by [7], the aim of the present paper is to introduce more general classes of endomorphisms and study their connection with commutativity of prime rings with involution of the second kind. Moreover, we will provide examples to prove that our results cannot be extended to semi-prime rings.

Endomorphisms with identity on commutator
We will use frequently the following facts which are very crucial for developing the proofs of our main results.
Proof. For the nontrivial implication, we are given that (2.1) Replacing x by x + y * in (2.1), we find that Replacing y by yh, where h ∈ Z(R) ∩ H(R)\{0} and using the last equation, we obtain In view of (2.1), we can see that [x, In light of the primeness of R, we have either Comparing Eqs (2.2) and (2.5), we may write And thus R is commutative by ( [3], Theorem 3). Assume that T (s) = −s for all s ∈ S(R) ∩ Z(R). Replacing y by ys in equation (2.2), where s ∈ Z(R) ∩ S(R)\{0}, we find that On In consequence of which T (x) + x * ∈ Z(R) for all x ∈ R, by Fact 1., this yields that As an application of our theorem, the following corollary improves the result of ( [1], Corollary 2) for the case when the underlying identity belongs to the center of a prime ring with involution of the second kind.
Proof. For the nontrivial implication, suppose that (2.10) Linearizing (2.10), one can see that Arguing as above we get T (z) ∈ Z(R) for all z ∈ Z(R). Multiplying (2.11) by T (h), and subtracting the result from equation (2.12), we conclude that (2.14) Using equations (2.11) together with (2.14), we conclude that In view of ( [8], Corollary 3.6.) the last equation implies that R is commutative or d u = 0. Hence the latter case yields to [u, v] = 0 for all u, v ∈ R and thus R is commutative integral domain. This completes the proof of our theorem.

Corollary 2.4. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind and T is an endomorphism of R, then R is a commutative integral domain if and only if T [x, y] + [x, y] ∈ Z(R)
for all x, y ∈ R.
The following example shows that the primeness hypothesis in Theorems 2.1 and 2.3 is not superfluous. In particular, our theorems cannot be extended to semi-prime rings. The following example proves that the condition "* is of the second kind" is necessary in Theorems 2.1 and 2.3.

Example 2.6. Let R = M 2 (Z) and let * be the involution of the first kind defined in Example 1. It is straightforward to check that all homomorphisms fulfilled the conditions of Theorems 2.1 and 2.3.
However R is not a commutative ring.

Endomorphisms with identity on anti-commutator
Our purpose in this section is to treat the commutativity of R in case the commutator in the preceding theorems is replaced by anti-commutator. Proof. For the nontrivial implication, assume that (3.1) Linearizing (3.1), we get Applying (3.1), we can see that x•x * , T (z) = 0 for all x ∈ R and z ∈ Z(R), then T (z) ∈ Z(R) for all z ∈ Z(R), by Fact 2. Using the same techniques as above, we get Using the primeness hypothesis, it follows that Replacing y by h in (3.5), where h ∈ Z(R) ∩ H(R)\{0}, it's obvious to verify that Replacing again y by s in (3.5), where s ∈ Z(R) ∩ S(R)\{0}, then x − x * ∈ Z(R) for all x, y ∈ R.
which reduces to Using equations (3.2) together with (3.9), we conclude that Replacing y in the last equation by a non zero element of Z(R) ∩ H(R) , we find that and thus R is a commutative integral domain by ( [3], Theorem 3).
Using equations (3.2) together with (3.12), we conclude that Replacing y by h in the last equation, where h ∈ Z(R) ∩ H(R)\{0}, we obtain (3.14) Since (3.14) is the same as (2.18). So, we may argue as before that R is a commutative integral domain.
A slight modification in the proof of Theorem 3.1 yields the following result. for all x ∈ R.
The following example proves that the condition "* is of the second kind" is necessary in Theorems 3.1 and 3.2. The following example proves that the primeness hypothesis in Theorems 3.1 and 3.2 is not superfluous. In particular, our theorems cannot be extended to semi-prime rings. Example 3.4. Let us consider (R, * ) as in the preceding example. Let (S, σ) be a commutative ring with involution of the second kind (for example the field of complex numbers with the conjugation involution). If we set R = R × S, then it is obvious to verify that (R, τ ) is a semi-prime ring with involution of the second kind where τ (r, s) = (r * , σ(s)) for all (r, s) ∈ R.
Furthermore, the zero endomorphism satisfies conditions of Theorems 3.1 and 3.2. But R is a noncommutative ring.