Commutativity of MA-Semirings with Involution through Generalized Derivations

Ali, Liaqat; Aslam, Muhammad; Qureshi, Muhammad Imran; Khan, Yaqoub Ahmed; Rehman, Shafiq Ur; Farid, Ghulam

doi:https://doi.org/10.1155/2020/8867247

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Research Article | Open Access

Volume 2020 | Article ID 8867247 | https://doi.org/10.1155/2020/8867247

Commutativity of MA-Semirings with Involution through Generalized Derivations

Liaqat Ali,¹Muhammad Aslam,²Muhammad Imran Qureshi,³Yaqoub Ahmed Khan,⁴Shafiq Ur Rehman,⁵and Ghulam Farid⁵

Academic Editor: Shaofang Hong

Received25 Aug 2020

Accepted21 Nov 2020

Published14 Dec 2020

Abstract

In this paper, we study the generalized derivations of MA-semirings with involution. We discuss some differential identities satisfied by the generalized derivations which force the semirings with involution to be commutative.

1. Introduction and Preliminaries

Javed et al. [1] introduced the notion of MA-semiring that is an additive inverse semiring satisfying condition of Bandlet and Petrich [2]. The notion of MA-semirings is groundbreaking to use commutators and their related identities in semirings. This enables the algebraists to produce and extend some remarkable results in this area. MA-semiring is a generalized structure of rings and distributive lattices but in spite of semirings we can deal with lie theory in MA-semirings. For ready reference, one can see [1, 3, 4]. The concept of commutators along with derivations and certain additive mappings was further investigated and extended in [1, 3–6]. Involution is one of the important and fundamental concepts studied in functional analysis and algebra. For instance, B∗-algebra due to Rickart [7] and C^∗-algebra due to Segal [8] are now well-known concepts that are defined with involution. Later on, many algebraists used this idea in groups, rings, and semirings (see [9–22]). Several research papers have been produced for MA-semirings with involution; for reference, one may see [5, 6]. To discuss the results of rings with involution in MA-semirings with involution would be of great interest for readers and researchers.

In this paragraph, we compose some necessary definitions and preliminary concepts. By a semiring , we mean a semiring with absorbing “0,” in which addition is commutative. A semiring is said to be additive inverse semring if for each there is a unique such that and , where denotes the pseudoinverse of . An additive inverse semiring is said to be an MA-semiring if it satisfies , where is the center of . In fact, every ring is MA-semiring, while converse may not be true. The following is one of the examples of MA-semiring which is not a ring. Such examples motivate us to generalize the results of ring theory in MA-semirings.

Example 1. (see [1]). The set with addition and multiplication , respectively, defined by and is an MA-semiring. In fact, is a commutative prime MA-semiring.
Throughout the paper, by we mean an MA-semiring unless stated otherwise. We say is prime if implies that or and semiprime if implies that . is 2-torsion free if, for , implies . An additive mapping is involution if, , and . An element is Hermitian if and skew Hermitian if . The set of Hermitian elements of is denoted by and that of skew Hermitian elements is denoted by .

Example 2. (see [5]). Let be an MA-semiring. Then the set with addition “+” and multiplication defined as forms an MA-semiring called the opposite MA-semiring of . We usually denote it as . Consider with and . Then forms an MA-semiring. Define by . Then defines an involution on . Thus, the triplet forms an MA-semiring with involution. We further see that MA-semiring is -prime but not prime. Therefore, this example also shows that a prime MA-semiring with involution is a -prime MA-semiring but converse is not true in general.
An additive mapping is a derivation if . The concept of generalized derivation was studied for MA-semirings in [6]. An additive mapping is a generalized derivation associated with a derivation if (see [23]). The commutator is defined as . By Jordan product, we mean for all . A mapping is commuting if , . A mapping is centralizing if , . The following identities are very useful for sequel: for all ,(1)(2)(3)(4)(5)(6)(7)For more details, one can see [1, 4].
In the following, we recall a few results for MA-semirings with involution, which are very useful for proving the main results.

Lemma 1 (see [24]). Let be a semiprime MA-semiring with involution of second kind. Then and therefore .

Remark 1. Let be an MA-semiring with involution of second kind.(1)For any , (2)For any and , Idrissi and Oukhtite [25] proved some results on generalized derivations satisfying certain conditions on rings with involution. The main objective of this paper is to prove the results for MA-semirings with involution.

2. Main Results

Lemma 2. Let be a 2-torsion free prime MA-semiring and let be a generalized derivation associated with a derivation of . If satisfiesthen is commutative.

Proof. By the hypothesis, for all , we have and thereforeIf , then (2) becomesIn (3), replacing by , we getFrom (3), we also have and therefore (4) becomesIn (5), replacing by and using (5), we obtainBy the primeness of , we get or . If , then is commutative. Secondly, ifit also impliesIn (7), replacing by , we get and, using (8), we obtainIn (9), replacing by and using (9) again, we get . As is prime and , . Therefore, is commutative.
We now consider the case when . In (2), replacing by and using (2) again, we havewhich further givesIn (10), replacing by , we obtainFrom (2), using in the last expression, we getand, using (10), we obtainUsing (11) into (14), we get and so . By the primeness of , we have either or . If , then , and therefore, by Theorem 2.2 of [26], is commutative. Secondly, if , then, replacing by , we obtain and hence . By the primeness of , we conclude that is commutative.

Lemma 3. Let be a 2-torsion free prime MA-semiring with involution of second kind. If satisfiesthen is commutative.

Proof. By the hypothesis, for all , we haveIn (16), replacing by , we get , which implies and therefore . By the 2-torsion freeness of , we have , which further means . In view of Lemma 1, . Therefore, by the primeness of , we obtain , which is further simplified to beIn (16), replacing by , we get and therefore . By the 2-torsion freeness of , we get and therefore . In the view of Lemma 1, ; therefore, by the primeness of , we have and henceand, using (17) into (18), we obtain which, by the 2-torsion freeness of , gives . This proves that is commutative.

Theorem 1. Let be a 2-torsion free prime MA-semiring with involution and let be a nonzero generalized derivation associated with a derivation of . If satisfiesthen is commutative.

Proof. Firstly, suppose that . Linearizing (19) and using (19) again, we getIn (20), replacing by , we getIn (21), replacing by , we get . As , , which further implies . In view of Lemma 1, by the primeness of , we get and thereforeUsing (22) into (21) and hence using 2-torsion freeness of , we get . and by Lemma 2 we conclude that is commutative.
Secondly, suppose that .
In (21), replacing by , we obtain and so . Rearranging the terms, we getUsing (21) into (23), we getIn (24), replacing by , we obtain and therefore . By the primeness , we have and thereforeUsing (23) into (24) and hence using 2-torsion freeness of , we get , which can be further written asIn (26), replacing by , we get and therefore . Rearranging the terms, . Using (26) again, we obtain . Therefore, we can easily obtain which, by the primeness, gives that either is commutative or . Suppose thatIn the view of Lemma 1, for any , ; therefore, replacing by in (27), we get , and, by the primeness of , we have . Replacing by in (21), we get and therefore . In the view of Lemma 1, using the primeness of , we have , which further impliesUsing (28) into (21) and then using 2-torsion freeness of , we obtain . Employing Lemma 2, we get the required result.

Proposition 1. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If andthen is commutative.

Proof. Suppose that . Define a mapping bywhere . We now prove that is generalized derivation.
Firstly, for any ,This shows that is additive. Secondly, for any ,This shows that is generalized derivation associated with the derivation . From (29), we can writeHence, by Theorem 1, we conclude that is commutative.

On the similar lines of Proposition 1, we can obtain the following proposition.

Proposition 2. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If andthen is commutative.

Theorem 2. Let be a 2-torsion free prime MA-semiring with involution . If is a nonzero generalized derivation satisfyingthen is commutative.

Proof. Linearizing (35) and using (35) again, we getAnd, replacing by , we further getSuppose that . In (37), replacing by , we get , which further gives . Therefore, in the view of Lemma 1, by the primeness of , we obtainand this impliesUsing (39) into (37) and then using 2-torsion freeness of , we obtainIn (40), replacing by , we obtain and therefore . In view of Lemma 1, by the primeness, we obtainIn (41), replacing by and using (41) again, we find and therefore . As , by the primeness, we have . This proves that is commutative.
Suppose that .
In (37), replacing by , we obtainand thereforeUsing (37) again, we obtainIn (44), replacing by , we obtainIn view of Lemma 1, by the primeness of , we haveand thereforeUsing (47) into (44) and the 2-torsion freeness of , we obtainIn (48), replacing by and using (48) again, we obtain and therefore , which further, by the primeness, implies that either is commutative orFrom (49), we can writeIn (49), replacing by , we getUsing (50), we getBy the primeness of , we have that either is commutative or . Suppose that . Following the same arguments as those in Theorem 1, we have , . In (37), replacing by and using , we obtain , which further impliesUsing (53) into (37) and then using 2-torsion freeness of , we getIn (54), replacing by and using the fact that , we obtain and hence, replacing by , we find . By Lemma 2, is commutative.

Proposition 3. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If andthen is commutative.

Proof. Define by , where . Then, following the same lines of the proof of Proposition 1, we find that is a generalized derivation satisfyingHence, by Theorem 2, is commutative.

On the similar lines, we can obtain the following result.

Proposition 4. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If andthen is commutative.

Lemma 4. Let be 2-torsion free prime MA-semiring with involution of second kind. Ifthen is commutative.

Proof. Linearizing (58) and using (58) again, we getand hence, replacing by , we further getIn (60), replacing by , we obtain and therefore . In view of Lemma 1, using the primeness of , we have and henceUsing (61) into (60), we obtain and since is 2-torsion free, Replacing by in (62) and using (62) again, we obtain and therefore , which, by the primeness, implies is commutative.

Lemma 5. Let be 2-torsion free prime MA-semiring with involution of second kind. Ifthen is commutative.

Proof. Linearizing (63) and using (63) again, we obtain and hence, replacing by , we obtainIn (64), replacing by , we obtain and therefore . In view of Lemma 1, we get and henceAs is 2-torsion free, using (65) into (64), we getIn (66), replacing by , we get and, employing Lemma 1, we conclude that is commutative.

Theorem 3. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If satisfiesthen is commutative.

Proof. When , (67) becomes . Therefore, employing Lemma 4, we conclude that is commutative.When , firstly suppose that . Linearizing (67) and using it again, we obtainAnd, replacing by , we obtainIn (69), replacing by and using the assumption , we obtainBy the primeness of , we getand henceAs is 2-torsion free, using (72) into (69), we getFrom (73), we can writeInterchanging and in (74), we obtain . But ; thereforeUsing (75) into (74), we get and, by the 2-torsion freeness of , we getIn (76), replacing by and using (76) again, we obtain and therefore . By the primeness of , we conclude that is commutative.
Next, suppose that .
In (69), replacing by and using (69) again, we obtainIn (77), replacing by , we obtain and therefore . In view of Lemma 1, by the primeness of , we obtain , which further givesAs is 2-torsion free, using (78) into (77), we getIn (79), replacing by and using (79), we obtain and therefore . By the primeness of , we get and therefore as above we conclude that either is commutative or and this further implies . In (69), replacing by and using the fact that , we obtain , which further implies . In view of Lemma 1, we haveIn (69), replacing by and using the fact that , we obtain , which further implies . In view of Lemma 1, we obtain and henceUsing (81) into (80) and then by the 2-torsion freeness of , we obtain . Following the same steps as Theorem 2 (equation (40)), we conclude that is commutative.

On the similar lines of Theorem 3, we can establish the following result.

Theorem 4. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If satisfiesthen is commutative.

Theorem 5. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . If satisfiesthen is commutative.

Proof. Suppose that . Then (83) becomes . By Lemma 5, is commutative.
Now suppose that . Consider the case when .
Linearizing (83) and using (83) again, we getIn (84), replacing by , we obtainIn (85), replacing by and using the assumption , we getand therefore, by rearrangement, we haveand henceIn view of Lemma 1, by the primeness of , we obtainwhich further givesUsing (90) into (85) and the 2-torsion freeness of , we obtainIn (85), interchanging and , we get . But since , , which implies . Hence,As is 2-torsion free, using (92) into (91), we findIn (93), replacing by and then using 2-torsion freeness of , we get and therefore . In view of Lemma 1, by the primeness of , we get . This shows that is commutative.
Next, suppose that .
In (85), replacing by , we getRearranging terms, we can writeUsing (85) again, we getIn (96), replacing by , we obtain , which further gives . As is prime, employing Lemma 1, we get and henceUsing (97) into (96) and hence using 2-torsion freeness of , we obtainEquation (98) is the same as (79) of Theorem 3; therefore, following the same steps, we conclude that either is commutative or . If , then .
Rearranging the terms of (85), we getIn (99), replacing by and using the fact that , we getwhich further impliesUsing (101) into (99) and then by the 2-torsion freeness of , we haveInterchanging and in (102), we obtain . As , the last equation becomes . This implies thatUsing (103) into (102), we get and by the 2-torsion freeness of , we haveIn (104), replacing by , we obtain and by the 2-torsion freeness of , it is implied that and therefore . In view of Lemma 1, by the primeness of , we conclude that is commutative.
In view of the above results, we can easily conclude the following results.

Theorem 6. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . Then the following statements are equivalent:(i)(ii)(iii)

Theorem 7. Let be 2-torsion free prime MA-semiring with involution of second kind and let be a generalized derivation associated with a derivation of . Then the following statements are equivalent:(i)(ii)(iii)

3. Concluding Remarks

Commutativity is a very important aspect of mathematics and is discussed in almost all of its branches. This article presents some results on generalized derivations of MA-semirings with involution of second kind. This research is useful for researchers who want to induce commutativity in semirings with additive mappings and opens the door for further research in this area. Other differential identities and different mappings can be studied to induce commutativity in semirings.

Data Availability

No data were used to support for this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally.

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Copyright © 2020 Liaqat Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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