On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

Abstract

Let be any arbitrary associative ring, a semiprime ideal, and a nonzero ideal of . In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.

1. Introduction

Throughout this study, represents an associative ring with the center . The symbols and , where , stand for the anticommutator and commutator , respectively. We will frequently use the basic commutator and anticommutator identities: , and , for all . A ring is said to be a prime ring if for all , implies or , and is said to be a semiprime ring if for all , implies . An ideal is said to be a prime ideal of if and for all , implies or , and is a semiprime ideal if and for any , implies . A mapping is said to be a -commuting mapping on if for all where . In particular, if , then is said to be a commuting mapping on if . Note that every commuting mapping is a -commuting mapping (put ), but the converse can be inaccurate in general (taking as a set of which has no zero such that , then is a -commuting mapping, but it is not a commuting mapping). An additive mapping is said to be a derivation of if holds for all . An additive mapping associated with a derivation is said to be a generalized derivation of if holds for all .

Daif [1] introduced the concept of multiplicative derivations, and it was motivated by the work of Martindale [2]. Dhara and Ali [3] have expanded the concept of multiplicative derivation to include multiplicative (generalized)-derivation. As a result, a mapping (not necessarily additive) is called multiplicative (generalized)-derivation if holds for all , where is any mapping (not necessarily a derivation nor an additive map). As a result, multiplicative (generalized)-derivation encompasses both multiplicative and multiplicative generalized derivation ideas. The ideas of multiplicative centralizers are covered by a multiplicative (generalized)-derivation associated with mapping (not necessarily additive). However, there are few articles on this subject (see [3–5] for a partial bibliography).

Every generalized derivation is obviously a multiplicative (generalized)-derivation on . However, the converse can be inaccurate in general. The following example shows this fact.

Example 1. Let be any ring and letWe define the maps and as follows: Then, it is simple to show that is a multiplicative (generalized)-derivation associated with a multiplicative derivation , but is not a generalized derivation of .
Numerous studies have revealed that the global structure of is typically related to the behavior of additive mappings formed on . Recent work has investigated the commutativity of the factor ring , where is the prime ideal of any arbitrary ring , using algebraic identities in , including derivations and generalized derivations (see [6–11]).
In 2021, Mamouni et al. [12] proved that any ring must be commutative or , where is a prime ideal of if admits a generalized derivation satisfying for all . In the same year, Rehman et al. [13] proved that any ring must be commutative or , where is a prime ideal of if admit generalized derivations and satisfying for all .
We investigate algebraic identities with multiplicative (generalized)-derivation involving semiprime ideal in this article without making any assumptions about semiprimeness on the ring in discussion. In addition, examples are given to show the need of the constraints imposed on the various theorems’ hypotheses.

2. The Main Results

Lemma 1. Let be a ring with a semiprime ideal and a nonzero ideal of . If such that , then . In particular, if such that or , then .

Proof. Assume that such that . That is, for all . Replacing by in the last relation, where , we have for all . That is, . By semiprimeness of , we obtain , as desired.
Now, if such that or , then , and so, similar to the earlier, we get , as desired.

Theorem 1. Let be a ring with a semiprime ideal and a nonzero ideal of and any mapping on . If and are multiplicative (generalized)-derivations of which satisfyfor all , then and are -commuting on . Moreover, if is an automorphism, then is commutative.

Proof. Assume thatReplacing by in (4) and using it, where , we haveLeft multiplying (5) by , where , we getTaking by in (5), we refer thatComparing (6) and (7) givesPutting by in (8), we obtainAgain, putting by in (8), we see thatThat is,Hence,Comparing (9) and (12), we find that . Putting by in the last relation, we haveRight multiplying the previous expression by and then replacing by , we getTaking by in the previous expression, we see that . Using Lemma 1 in the last relation, we get . Taking by in the last relation and using it, where , we get . It is clear to see that . In particular, . By using Lemma 1 in the last relation, we obtainas desired.
Now, right multiplying (8) by and then replacing by and then subtracting them, we getUsing (15) in the last relation, we . Taking in the last relation and using Lemma 1, we see thatas desired.
Now, assume that is an automorphism. By using (15) and (17) in (7), we obtain . Since is an automorphism, we get . Since is a semiprime ideal, we see thatThat is,Hence,Using Lemma 1 in the previous expression, we infer thatReplacing by in (21) and using it, where , we obtain . Taking by in the previous expression and using it, we have . It follows that . Using (21) in the last relation, we get . Putting by in the previous expression and using it, we see that . Using Lemma 1 in the previous expression, we conclude that . Replacing by in the previous expression and using it, where , we obtain . Again, using Lemma 1 in the last relation, we have . Similarly, from the last relation, we can get for all . It follows that is commutative.

Corollary 1. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfyingfor all , then is commutative.

Proof. Since is a multiplicative (generalized)-derivation, we get (where 1 is an identity map) is a multiplicative (generalized)-derivation. Now, put and in Theorem 1.

Theorem 2. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)for all , then is commutative.

Proof. (1)Putting in Theorem 1, we get the desired result.(2)Assume thatTaking by in the previous expression and using it, where , we haveNow, since is any mapping in Theorem 1, we can put in (5), and then using the same technique, we get (15) and (17), and by using both equations in (24), we obtain (18), and so we get the desired result.

Corollary 2. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)for all , then and are -commuting on .

Proof. (1)Since is any mapping in Theorem 1, we can put , and so we get the desired result.(2)Putting by in (5), we obtain (4).

Theorem 3. Let be a ring with a semiprime ideal and a nonzero ideal of and any mapping on . If and are multiplicative (generalized)-derivations of satisfyingthen is a -commuting on .

Proof. Assume thatReplacing by in (26), where , we haveBy using (26) in (27), we getPutting by in (28) and then left multiplying it by and then subtracting them, we see thatTaking by in (29) and then right multiplying it by and then subtracting them, we find thatThat is,Hence,It follows thatReplacing by in the last relation and then by in it and then subtracting them, we infer thatThat is,Putting by in the last relation and then left multiplying it by and then subtracting them, where , we have . Hence, . Taking by in the last relation, we get . Using Lemma 1 in the last relation, we getTaking by in (36), where , we see that . Using (36) in the last relation, we find that . Putting by in the last relation, we getUsing Lemma 1 in the last relation, we find that . Using the last relation in (36), we get . It follows that . Using Lemma 1 in the last relation, we see that for all .
When and in Theorems 1 and 3, we have the following corollary.

Corollary 3. Let be a ring with a semiprime ideal and a nonzero ideal of . If is a multiplicative (generalized)-derivation of satisfying any one of the conditions(1)(2)for all , then is a -commuting on .

Theorem 4. Let be a ring with a semiprime ideal and a nonzero ideal of . If is a multiplicative (generalized)-derivation of satisfying for all , then is a -commuting on .

Proof. (1)Assume thatPutting by in (38), we haveand by using (39) in the last relation, we getReplacing by in (40), we obtainLeft multiplying by in (40) and then subtracting from (41), we see thatAgain, replacing by in the last relation and then right it by and then subtracting them, we infer thatUsing Lemma 1 in the last relation, we get for all .

Theorem 5. Let be a ring with a semiprime ideal and a nonzero ideal of . If is a multiplicative (generalized)-derivation of satisfying for all , then is a -commuting on .

Proof. Putting by in our hypothesis, we get is a -commuting on .

Theorem 6. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)(3)(4)for all , then (moreover, for all ), and and are -commuting on .

Proof. (1)Assume that Taking by in (44), where , we have By using (44) in the last relation, we get Putting in (46) and then left multiplying it and then subtracting them, where , we obtain . Thus, . Taking in the last relation, we conclude . Using Lemma 1 in the last relation, we get . That is, as desired. Taking by in (47) and using it, where , we get . That is, From (46), we get . Putting by in the previous expression and then right multiplying it by and then subtracting them, where , we obtain Using (48) in the last relation, we see that . Taking by in the last relation, where , we find that . That is, . Replacing by in the previous expression and using it, we infer that . Again, replacing by in the previous expression and using it, where , we conclude that . Putting and in the previous expression, where , we have . Using Lemma 1 in the previous expression, we obtain In particular, for all , as desired. Putting in (50) and using it, we get . Replacing by in the previous expression and using it, we see that . It follows that . Using Lemma 1 in the previous expression, we infer that for all , as desired.(2)Assume that Replacing instead of in (51), where , we have By using (51) in (52), we get Now, similar to (46), we conclude the desired result.From (6) and (7), since is a multiplicative (generalized)-derivation, we infer that (where 1 is an identity map) is a multiplicative (generalized)-derivation; now putting in (44), we get the desired result.

Corollary 4. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)(3)(4)(5)(6)for all , then (moreover, for all ), and and are -commuting on .

Proof. Taking by and by in (1) and then using Theorem 6 (1), we get (1). Taking by in (2), we get (1), and then using the previous trick, we obtain (2). Taking by and by in (3) and then using Theorem 6 (2), we get (3). Taking by in (4) and then using (3), we get (4). Taking by in (5) and then using (3), we get (5). Taking by and by in (6) and then using (4), we get (6)..

Theorem 7. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)(3)(4)(5)for all , then and for all .

Proof. (1)Assume thatReplacing by in (54) and using it, we havePutting by in (55) and then left multiplying it by , where , and then subtracting them, we getTaking by in (55) and then left multiplying it by and using (56), we see thatAgain, taking in (57), this givesand so . Using Lemma 1 in the last relation, we getas desired.
Putting by in (57), where , we getand so . In particular, . Now, right multiplying (55) by and using the last relation and (59), we get . Left multiplying the last relation by , we obtain . Using Lemma 1 in the previous expression, we see that for all , as desired.
(5)–(8) are similar to (55).

Theorem 8. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfyingfor all , then and for all .

Proof. Assume thatReplacing by in (62), we haveBy using (62) in the last relation, we getNow, similar to the proof in Theorem 7(1), equation (55), we get and for all .
From Theorem 8, we get the next result.

Corollary 5. Let be a ring with a semiprime ideal and a nonzero ideal of . If and are multiplicative (generalized)-derivations of satisfying any one of the conditions(1)(2)(3)(4)for all , then and for all .

Example 2. Let and is an identity mapping on . We define asIt is easy to verify that and are multiplicative (generalized)-derivations on associated with the mapping and on , respectively. Letand . Here, we see that is an ideal of satisfying the following conditions:(1)(2)(3)for all but is noncommutative. Note that but , and so is not a semiprime ideal. Hence, the semiprimeness of our hypothesis is essential in Theorem 2 and Corollary 4.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.