ENDOMORPHISMS WITH CENTRAL VALUES ON PRIME RINGS WITH INVOLUTION L. Oukhtite, H. EL Mir and B. Nejjar

In this paper we present some commutativity theorems for prime rings R with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25


Introduction
Throughout this article, R will represent 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 anti-commutator xy + yx. R is 2-torsion free if whenever 2x = 0, with x ∈ R implies x = 0. R is prime if aRb = 0 implies a = 0 or b = 0. An additive mapping * : R −→ R is called an involution if * is an anti-automorphism of order 2; that is (x * ) * = x and (xy) * = y * x * for all x, y ∈ R. 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 skewhermitian elements of R will be denote 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 latter case S(R) ∩ Z(R) = {0}. A derivation on R is an additive mapping d : R −→ R such that d(xy) = d(x)y + xd(y) for all x, y ∈ R. In [10], Brešar introduced the notion of generalized derivations in rings: an additive mapping F : R −→ R is called generalized derivation if there exists a derivation d such that F (xy) = F (x)y + xd(y) holds for all x, y ∈ R, and d is called the associated derivation of F. Over the last 30 years, several authors have investigated the relationship between the commutativity of the ring R and certain special types of mappings on R. The first result in this direction is due to Divinsky [14], who proved that the simple artinian ring must be commutative if it has a commuting nontrivial automorphism. Two years later, Posner [21] proved that the existence of nonzero centralizing derivation on prime ring forces the ring to be commutative.
We say that a map f : The study of commutativity preserving mappings has been an active research area in matrix theory, operator theory and ring theory (see [9], [22] for references). A map f : R −→ R is said to be strong x, y in a right ideal I of R, then I ⊆ Z(R). In particular, R is commutative if I = R. Later, Deng and Ashraf [13] proved that if there exist a derivation d of a semiprime ring R and a map f : for all x, y ∈ I, then R contains a nonzero central ideal. In particular, they showed that R is commutative if I = R. Further, Ali and Huang [1] showed that if R is a 2−torsion free semiprime ring an d is a derivation of R satisfying [d(x), d(y)] + [x, y] = 0 for all x, y in a nonzero ideal I of R, then R contains a nonzero central ideal. Many related generalizations of these results can be found in the literature (see for instance [2], [5], [7], [11], [12], [15], [17]).
Our aim in the present paper is to continue this line of investigation that consists on the study some special endomorphisms and their connection with commutativity for prime rings.

Commutativity and P i -endomorphisms
In [16] it is proved that the prime ring R must be commutative if R is equipped with a generalized derivation F associated with a nonzero derivation d satisfying any one of the following conditions: for all x, y ∈ I, where I is a nonzero two sided ideal of R.
Motivated by the previous results, our purpose in this section is to examine what happens in case the generalized derivation F is replaced by an endomorphism g satisfying any one of the following properties: More precisely, we will discuss existence of such mappings and their relationship with commutativity in case of prime rings.
Definition 2.1. Let R be a ring and g : R −→ R an endomorphism. For i ∈ {1, 2, 3, 4} we will say that g is a P i -endomorphism if g satisfies the property P i . Remark 2.2. If R is a commutative ring, then every endomorphism is a P i - Our next aim is to give examples of P i -endomorphisms over noncommutative rings.
then it is obvious to verify that the endomorphism g defined on the noncommutative ring R by: where α ∈ Z, is a P i -endomorphism for i ∈ {1, 2, 3, 4}.
(2) Let S = R × C where C is a commutative ring and R is the ring provided with the endomorphism g defined in the example (1). It is straightforward to check that where h is any endomorphism defined on C, is a P i -endomorphism for i ∈ {1, 2, 3, 4}.
The following theorem gives some commutativity criteria through the behavior of the P i -endomorphisms.
Theorem 2.4. Let R be a prime ring and g an endomorphism of R. The following assertions are equivalent: (1) g is a P 1 -endomorphism; (2) g is a P 2 -endomorphism; (3) g is a P 3 -endomorphism; (4) g is a P 4 -endomorphism; Proof. For the non-trivial implications assume that R is not commutative.
(1) ⇒ (5) We are given that Substituting yx for y in equation (2) one can see that in such a way that [ Using the primeness of R together with Brauer's trick (that is a group cannot be a union of two of its proper subgroups), we conclude that g = I d . Hence (2)  invoking the primeness of R we get [x, y] = 0 for all x, y ∈ R which contradicts the fact that R is not commutative. Hence g(z 2 ) = z 2 , and thus taking y = z 2 in the hypothesis, we obtain which, as above, contradicts the fact that R is not commutative. Then, we conclude that R is a commutative integral domain.
(2) ⇒ (5) Follows from the first implication with a slight modification.

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(3) ⇒ (5) Suppose that If Z(R) = {0}, then our assumption becomes Substituting yx for y in equation (6), we arrive at replacing y by yr, we obviously get By view of equation (7), equation (8) yields Since R is not commutative then g = I d and the hypothesis becomes and thus R = {0}, a contradiction. Hence Z(R) = {0}; replacing y by yz 2 in Since R is not commutative, then g(z 2 ) = z 2 ; replacing y by z 2 in the hypothesis, we find that substituting xy for x we obtain Using the last equation together with equation (5) one can see that hence R is commutative, a contradiction. We conclude that R is a commutative integral domain.
The following example proves that the primeness hypothesis of R is necessary in R is not prime, because of XY Z = 0 for all X, Y, Z ∈ R. Moreover, for all X, Y ∈ R we have [X, Y ] ∈ Z(R) and X • Y ∈ Z(R). Hence the zero endomorphism Θ R is a P i -endomorphism for i ∈ {1, 2, 3, 4} but R is not commutative.

Commutativity and P * i -endomorphisms
The authors in [16] explore the commutativity of a prime ring with involution (R, * ) provided with a generalized derivation F satisfying any one of the following conditions: for all x ∈ R. Our purpose in this section is to continue this line of investigation by studying commutativity criteria for rings with involution admitting an endomorphism satisfying certain algebraic identities in a more general situation. More precisely, we will establish an endomorphism version of the above properties.
Therefore, we need to introduce more general classes of mappings as in the following definition.
Definition 3.1. Let (R, * ) be a ring with involution and let g : R −→ R be an endomorphism. Then g is called a: Since [X, X * ] = 0 and X • X * ∈ Z(R) for all X ∈ R, then the zero function Θ R is a P * i -endomorphism for i ∈ {1, 2, 3, 4}. Also, if we take g : R −→ R the endomorphism defined by g(X) = P XP −1 with P = 3 0 0 1 , then for all X ∈ R we have g(XX * ) = XX * = X * X ∈ Z(R) so that g is a P * i -endomorphism for i ∈ {1, 2, 3, 4}.   We are not able to find an example of a P * i -endomorphism which is not a P iendomorphism in the case where (R, * ) is a prime ring with involution of the second kind. Consequence of which it's natural to consider the following conjecture: Conjecture: Let (R, * ) be a 2-torsion free prime ring with involution of the second kind and g an endomorphism of R. Then g is a P * i -endomorphism if and only if g is a P i -endomorphism for i ∈ {1, 2, 3, 4}.
The next theorem gives an affirmative answer to the above conjecture. However, we will use frequently the following Lemma which is very crucial for developing the proofs of our theorem.   Proof. (i) Let a ∈ R, we are given that Linearizing this equation and replacing y by y * , we find that Substituting ys for y, where s ∈ S(R) ∩ Z(R), and using the last equation, we obtain [[x, y], a] = 0 for all x, y ∈ R.
Replacing y by yx one can see that Since R is semiprime we deduce that [x, a] = 0 for all x ∈ R, hence a ∈ Z(R).
(ii) Let a ∈ R, we are given that Linearizing this equation and replacing y by y * , we find that Substituting ys for y and using (18), we obtain Taking y = z where z ∈ Z(R)\{0} we arrive at Then we conclude that a ∈ Z(R).
Theorem 3.5. Let (R, * ) be a 2-torsion free prime ring with involution of the second kind. If g is a non-trivial endomorphism of R, then g is a P * i -endomorphism if and only if g is a P i -endomorphism for i ∈ {1, 2, 3, 4}.
Proof. For the non-trivial implications, we need to prove that g is a P * i -endomorphism implies that g is a P i -endomorphism for i ∈ {1, 2, 3, 4}.
1) Assume that g is a P * 1 -endomorphism, then Linearizing equation (21) and replacing y by y * , we find that Replacing y by yh in (22), where h ∈ Z(R) ∩ H(R), we obtain Using equation (21), where h ∈ Z(R) ∩ H(R), one can see that Substituting x * for y we get Invoking the primeness of R, the last relation implies that either [x, for all x ∈ R, we deduce that R is commutative and therefore g is a P 1 -endomorphism. If g(h) = h for all h ∈ Z(R) ∩ H(R) then g(s) = s or g(s) = −s for all s ∈ Z(R) ∩ S(R).
Assume that g(s) = s; replacing y by ys in equation (22) we find that Using equation (26) together with equation (22), it follows that for all x, y ∈ R and thus g is a P 1 -endomorphism.
If g(s) = −s for all s ∈ S(R) ∩ Z(R); substituting ys for y in equation (22)  Substituting s for y, where s ∈ Z(R) ∩ S(R), one can see that thereby, the relation (21) gives [x, x * ] ∈ Z(R) for all x ∈ R and using [18, Lemma 2.1] we conclude that R is commutative, then g is a P 1 -endomorphism.
2) Assume that g is a P * 2 -endomorphism. Using similar argument as in the proof of 1) we can prove that g is a P 2 -endomorphism.
3) Suppose g is a P * 3 -endomorphism, that is Then a linearization of (29) forces Writing yh instead of y in the last equation, where h ∈ Z(R) ∩ H(R), we obtain By view of Lemma 3.4, equation (30) yields Subtracting (31) from (32), we find that Substituting x * for y we get .
for all x ∈ R then R is commutative by [18, Lemma 2.2], hence g is a P 3 -endomorphism. If g(h) = h then g(s) = s or g(s) = −s for all s ∈ Z(R)∩S(R).
Suppose that g(s) = s, then replacing y by ys in equation (30) we find that Comparing equations (30) and (35), we are forced to conclude that g(x)g(y) − x • y ∈ Z(R) for all x, y ∈ R hence g is a P 3 -endomorphism.
Putting s instead of y, where s ∈ Z(R) ∩ S(R), one can see that

4)
Assume that g is a P * 4 -endomorphism. Using similar argument as in the proof of 3) we can prove that g is a P 4 -endomorphism.
The following example proves that the condition "* is of the second kind" is necessary in Theorem 3.5. is straightforward to check that (R, * ) is a prime ring and * is an involution of the first kind. Moreover, for all X ∈ R, we have [X, X * ] = 0 and X • X * ∈ Z(R).
In this second example we will show that Theorem 3.5 cannot be extended to semiprime rings.