Higher Derivations Satisfying Certain Identities in Rings

Abstract

Let and be fixed positive integers. In this paper, we establish some structural properties of prime rings equipped with higher derivations. Motivated by the works of Herstein and Bell-Daif, we characterize rings with higher derivations satisfying (i) for all and (ii) for all .

1. Introduction

Knowing the structure of a ring has always been an interest to various authors, be it its number of ideals, commutativity, and behavior for certain special identities. In this pursuit, derivations were often used. Like Posner, Herstein, Bell, Daif, and a number of other enthusiastic mathematicians brought the idea of differential identities to study the structure of rings. There are some other types of derivations such as Jordan derivations, derivations, generalized derivations, and skew derivations which can be proved helpful for the same. Indeed, most of these forms of derivations have already proven fruitful. The usage of the idea of higher derivations in this direction is relatively a new one. To explore this further, we first revisit the idea of higher derivations on rings. An additive map is said to form a derivation if for all where is any ring. Let be the set of all nonnegative integers. Following [1], a family of additive mappings from to itself with defined as the identity map on is termed as a higher derivation if for all and . On similar lines, a higher derivation of rank is defined as a family of additive mappings (say) , where is the identity map on and for all and (see [2], page 540 for more details). A ring is prime if implies either or and semiprime if implies . The center of is denoted by . will denote the Martindale ring of quotients of and will represent its extended centroid. Recall that a derivation is said to be inner if there exists such that , for every . This derivation is usually called the inner derivation associated with ’ and will be denoted here by . Also, if is a derivation of and there exists such that , for every , then is said to be -inner [3]. In this case, we will denote again by .

The main aim of this paper is to describe the structure of prime rings equipped with higher derivations.

2. Background

Let us mention some known results which will be used frequently while proving our results:

Lemma 1 (see [4]). Let be a prime ring with 1 and let . Suppose for all . Then, either is -dependent or is -dependent.

Lemma 2 (see [1]). Assume that is a prime ring and is a higher derivation of which satisfies an -linear relation of minimal length on . Then there exist such that , for every .

Lemma 3 (see [5]). Let be a ring without any nonzero nilpotent ideals. Then any element of which commutes with all elements of must lie in the center of .

Lemma 4 (see [1]). (Proposition 1.1) Letbe a semiprime ring anda higher derivation of. Then there exists a unique higher derivationofsuch that, for every.

Fact 5 (see [6]). and satisfy the same generalized polynomial identity (GPI).

Lemma 6. Let be any ring and be a higher derivation defined on , where . Then whenever .

Proof. Clearly, the result holds for . For , we haveThis givesTherefore, . Thus, the result is true for . Let us suppose the result is true for , that is, for all . For , we haveOn computing the above equation, we obtainUsing the principle of mathematical induction, we conclude that for all . Hence, this is the desired result.

3. Herstein’s Result for Higher Derivations

Herstein [7] in the year 1978, proved that for a prime ring with a nonzero derivation satisfying for all , is a commutative integral domain if , and for , is commutative or is an order in a simple algebra which is 4-dimensional over its center. This result was extended to semiprime rings in 1998 by Daif [8], and furthermore, it was extended for any two-sided ideal of by Bell and Daif [9]. Bell and Rehman [10] applied this identity for generalized derivations and were able to obtain some interesting results. Motivated by these ideas, we wish to explore similar criteria for the case of higher derivations and describe the structure of prime rings. In fact, we prove the following result:

Theorem 7. Let be a prime ring and , be fixed positive integers. Next, let be a nonzero higher derivation of , where satisfying for all . Then, either some linear combination of maps center of to 0 or is commutative.

Proof. For a fixed pair , we have for all . Replace by , where to obtainOn solving, we obtainfor all , which givesOn commuting with , we obtainfor all , . Replace by , , to obtain which gives. On computation, we get. By Fact 5, and , its Martindale ring of quotients, satisfy the same generalized polynomial identity, and every higher derivation on can be uniquely extended to a higher derivation on by Lemma 4. Therefore, we conclude thatfor all , . The application of Lemma 1 helps us deduce that, either the set is linearly dependent over or the set is linearly dependent over . Suppose the set is linearly dependent over . So, there exist scalars from not all zeros such that,Hence, we have, , proving our claim that some linear combination of ’s maps center of to zero. Next, we suppose the set is linearly dependent over . There exist scalars from not all zeros, such thatLet be the highest index such that , making the sumNow, set the inner derivations asTherefore, the set satisfies a -linear relation over of length . The application of Lemma 2 yields that there exist elements such thatIn addition to this, andUsing , we obtainReplace by to obtainHence, we obtain, . Next, by using (18), we havefor all , . Replacing by , where ,Using the fact that for all , we obtainTherefore, for all . Hence, (18) becomesfor all and . For , the above equation becomesfor all . Subtracting (25) from (17), we obtainthat is,Since , we have for all . So, we can conclude for all . Thus, we havewhich impliesTherefore,for all . Replacing by , we obtainfor all . Sincewe obtainwhich impliesorfor all . Since is prime, thereforeThus, for all . Since forms a subring of , we conclude for all . Hence, for all . The application of ([12], Theorem 3.1) implies either is commutative or some linear combination of ’s maps center to zero. Hence, this is the required result.

Corollary 8. Let be a prime ring. Next, let be a derivation on such that for all , then either or is a commutative ring.

Proof. Take rank of the higher derivation one, i.e., in Theorem 7 and put . So, we have a family of . The application of Theorem 7 gives either is commutative or some linear combination of maps center of to zero, i.e., for all . This implies for all . Since was arbitrary, we obtain .

4. Bell’s Result for Higher Derivations

In the year 1998, Bell and Daif [9] while extending Herstein’s results proved a very interesting idea. With a derivation defined on a prime ring , he showed that if for all in a two-sided ideal of , then the ring is commutative. It was further extended to generalized derivations by Ali et al. in [11]. Motivated by this, we plan to venture this identity in case of higher derivations.

Theorem 9. Let be a fixed positive integer and be a prime ring. Next, let be a nonzero higher derivation of , where satisfying for all . Then, either is commutative or some linear combination of maps center of to 0.

Proof. We havewhich impliesfor all . On further solving, we obtainfor all . Now, replacing by , where , we obtainSince , the above equation becomesIn view of (39), we obtainfor all , . On solving this further, we obtainfor all , . For any , we havefor all , . Replacing by in the above equation, we obtainfor all , . This implies thatfor all , . By Fact 5, and , its Martindale ring of quotients, satisfy the same generalized polynomial identity and every higher derivation on can be uniquely extended to a higher derivation on by Lemma 4. Therefore, we deduce thatAs , for a prime ring is prime and , the application of Lemma 1 gives either the set is linearly dependent over or the set is linearly dependent over . Suppose first, that the set is linearly dependent over . So, there exist scalars from not all zeros, such thatSince is a subring of , therefore we can rewrite the above equation asthus proving our claim that some linear combination of maps center of to 0.
Next, suppose the set is linearly dependent over . Again, we will have scalars not all zeros from , such thatSuppose is the highest index such that . Thus, the sum becomesNow, set the inner derivations asSo, the set of derivations satisfies a -linear relation over of length . In view of Lemma 2, we have the existence of , such thatIn addition to this, andfor all , . With , we obtaingiving usReplacing by , we obtainTherefore, . Also,for all , . That isfor all . Replacing by in (59) and using the fact that , for all , the above equation givesThus, for all . So, for , (59) reduces toSubtracting (61) from (53), we obtainTherefore,Since , we haveThe application of Lemma 3 gives . As was chosen arbitrarily from , we conclude that is commutative. being a subring of must be commutative as well. This completes the proof of the theorem.

Corollary 10. Let be a prime ring. Next, let be a derivation on such that for all , then either or is a commutative ring.

Proof. Take rank of the higher derivation one, i.e., in Theorem 9 and put . So, we have a family of . The application of Theorem 9 gives either is commutative or some linear combination of maps center of to zero, i.e., for all . This implies for all . Since was arbitrary, we obtain .

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.

Acknowledgments

The authors extend their appreciation to Princess Nourah Bint Abdulrahman University (PNU), Riyadh, Saudi Arabia, for funding this research under Researchers Supporting Project Number (PNURSP2024R231).