Abstract
We introduce the notion of -homoderivation on a ring and show that a semiprime ring must have a nontrivial central ideal if it admits an appropriate -homoderivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.
1. Introduction
Studying the properties of rings equipped with mappings is an idea that began in the second half of the last century. Researchers have focused on two types of mappings defined on rings, namely, homomorphisms and derivations. Many authors studied the properties of prime and semiprime rings with derivations, such as commutativity of rings. It has inspired many researchers to generalize this mapping from various sides and study the commutativity of the rings and other properties, see [1–4]. At the same time, they dealt with the study of rings with homomorphisms and studied properties similar to those considered in the previous case, see [5–9].
With the beginning of the new millennium, the idea of combining the two mappings into one new mapping called homoderivation arose, and several different concepts emerged from this idea.
In [10], El-Sofy presented the first concept of homoderivation as a linear mapping that verifies, for all in the ring, the property . He gave examples showing the existence of such map. He showed that this map is neither a derivation nor a homomorphism. Also, he gave some results concerning the commutativity of some types of rings endowed with homoderivation and studied other properties of rings. Recently, many authors have studied and generalized the concept of homoderivation on different types of rings and near rings in several ways, and they also studiedthe properties of rings equipped with homoderivations, see [11, 12].
In [13], M. Mahdi Ebrahimi and P. A. JooHesh presented the second concept, which is completely different from the first one, as a positive linear mapping defined on the -ring and fulfilled the two conditions and and called that mapping homoderivation. Also, they defined antihomoderivation to be a derivation that is antihomomorphism. They generalized some results of Bell and Kappe [14].
As a generalization of El-Sofy’s homoderivation in 2020, Eszter Gselmann and Gergely Kiss defined the third concept, called -homoderivation, as a mapping on a ring and satisfying, for all in the ring, the condition , where belongs to the center of the ring. They characterized this mapping without additivity supposition. Obviously, if a ring has a zero center, then -homoderivation becomes a derivation. El-Sofy’s homoderivation can be considered as a special case of -homoderivation if the ring is unital. Also, they proved some results concerned with Mahdi’s homoderivation, see [15].
In this paper, we introduce the new concept, -homoderivation, as follows:
Definition 1. Let be a ring. An -homoderivation is a linear mapping which satisfies, for all and , , where is an integer.
Clearly, derivations can be considered as 0-homoderivation, the identity mapping is -homoderivation, the zero mapping is -homoderivation for all integer , and El-Sofy’s homoderivation can be considered as 1-homoderivation. So, the class of -homoderivations is wide rather that the notion of homoderivations and derivations. Although in rings with unity the concept of -homoderivation can be considered as a special case of -homoderivation, but in general, that is not true as shown in following example.
Example 1. The identity map on any nonunital ring is -homoderivation with , but it is neither homoderivation nor derivation. Also, there is no central element which makes the identity mapping -homoderivation. Therefore, the classes of -homoderivation and -homoderivation are different.
The following theorem shows that the notion of -homoderivation can be characterized in terms of homomorphism.
Theorem 1. Let be a ring and assume that is an arbitrary nonzero integer. If a linear map fulfills the equationthen there exists a homomorphism such that for all . Moreover, .
Proof. Clearly, since is linear, is additive. Multiplying equation (1) with leads to for all . If we add to both sides of this equation, then for all . Observe, however, that for all and yielding exactly that the mapping defined through for all is a homomorphism. Now, since for all , this completes our proof.
We give a generalization of a well-known result due to Bell and Martindale’s study [5] which stated that a semiprime ring must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Also, they proved commutativity in prime rings under some conditions.
Throughout this work, will be an associative ring with center , and will be an -homoderivation.
2. Results
To get our results, we will use the following facts about prime rings.
Fact 1. In a prime ring , the centralizer of any nonzero one-sided ideal is equal to the center of ; in particular, if has a nonzero central ideal, must be commutative.
Fact 2. Let be a left ideal in a semiprime ring . If is an epimorphism on such that for some , then .
Proof. Suppose , then . Therefore, .
The following lemma plays a crucial role in our present work.
Lemma 1. Let be a semiprime ring and be a nonzero left ideal. If is a centralizing -homoderivation on , then is commuting on .
Proof. For an arbitrary , we haveTherefore, we getThat is,Which givesAnd again, this givesBut , hence we getCommuting (7) with , we getSo, we haveand this givesAfter collecting, (10) can be reduced toConsequently, we getBut by using (12), we haveTherefore, we getBut is central and nilpotent, as well as, is semiprime, henceMoreover,that is,Linearizing in (15), we getAlso, since , thenNow, using (15), (18), and (19), we getThat is,Replacing by in (21), we getUsing (17), we getThat is,Therefore, using (21), we haveThus,In (26), replacing by , we haveTherefore, using (15) in (27), we getso,Now, if is even, then for some , so from (18), we have , henceIn (30), substituting instead of , we getNow, by using (15), (17), (31), and , we getButHence, for all ; that is, is commuting on for every even integer .
Now, if is an odd integer, then there exists such that , and then (29) can be rewritten asSo from (18), we have ; hence, (34) becomesIn (35), substituting instead of , we getNow, by using (15), (17), (36), and we getNow, (37) can be rewritten asIn (21), replacing by , we getThat is,Using (17) in (40), the third term is zero, and using , we can getBut using (15), we getSo we getComparing (41) and (43), we getNow, using (44) in (38), we getNow, we haveBut is central and nilpotent, as well as, is semiprime, henceThat is, is commuting on for every odd integer . Actually, is commuting on for every integer .
Lemma 2. Let be a ring, be an -homoderivation, and be a left ideal on . If for some , then(1)(2).(3)(4)Moreover, if , then
Proof. Suppose . Then, for any , we have(1).(2). Therefore, if is a left invariant ideal in by , then it is invariant by .(3).(4).
Theorem 2. Let be a nonzero left ideal in a semiprime ring , then admits an -homoderivation and centralizing on , and the corresponding homomorphism is epimorphism. If there exists an element such that , , and then contains a nonzero central ideal.
Proof. By Lemma 1, since is commuting on , we getReplacing by in (48), we get , which may be written asSubstituting for in (49) and using (49), we obtainSince is a left ideal, we getThe ring is semiprime; hence, we can choose a family of prime ideals of for which . From the last relation, it follows that for each , either , for all or A prime ideal is called a type-one prime if it satisfies and a type-two prime otherwise; let and be, respectively, the intersections of all type-one and type-two primes. Note that .
We now investigate a typical type-two prime . From , we have , but ; hence, for all . Consequently, for all , and so, for all . Putting , we get , but ; hence, for all . Since is a left ideal, then for all . The fact that is prime yields or . But if the latter holds true, then holds true for , contradicting our definition of a type-two prime ideal; therefore, . Thus, for all , so we have for all . By the condition of , we have , and by , we get . But , hence we have for all . Therefore, for all by primness of or . The latter case leads to a contradiction; therefore, , that is, . Consequently, . For any and , we have , that is, , which gives for all and . Thus, . But , hence . Since is a typical type-two prime, thenSince is an epimorphism, then as and , by Fact 2, are left ideals in , where is the left ideal generated by . Now, we claim that . To prove our claim, let . by Theorem 1. On the other side, , by Lemma 2. Hence, .
Consider now the left ideal generated by the set . We shall show that is commutative and hence a two-sided central ideal. A typical element of is a sum of elements of the form and , where and . Thus, we need to only show that commutators of the forms , , and are all trivial, where , , , and . Clearly, all three types are in by , and they are all in by (52); hence all belong to .
If , we are finished. Therefore, we assume that , in which case . The left ideal is therefore nilpotent, so . Thus, for all and all so that , and therefore, we getSince , then . So, we get for all and hence, we get . This contradicts our initial hypothesis, so the central ideal must in fact be nonzero.
Using Fact 1, the following theorem comes from Theorem 2.
Theorem 3. Let be a prime ring and be a nonzero left ideal. Suppose admits an -homoderivation which is centralizing on and the corresponding homomorphism is epimorphism. If there exists an element such that , , and , then is commutative.
As a special case of our work when , the corresponding homomorphism will be the identity map which will be an epimorphism. Also, there exists an element such that and . Also, since is centralizing and hence commuting; therefore, . Hence, . Also, . So, if , then we get the following results.
Corollary 1. Let be a semiprime ring and be a nonzero left ideal. If admits a 0-homoderivation (i.e., derivation) which is centralizing on and there exists an element , such that , then contains a nonzero central ideal.
Corollary 2. Let be a prime ring and a nonzero left ideal. If admits a 0-homoderivation (i.e., derivation) which is centralizing on and there exists an element , such that , then is commutative.
Data Availability
No data were used to support the findings of this study.
Disclosure
Current address for A. Ageeb and A. Ghareeb is Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65799, Saudi Arabia.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are indebted to Professor M. N. Daif, Al-Azhar University for his helpful suggestions and valuable comments which helped in constituting the paper in its present form.