Commutativity of MA-Semirings with Involution through Generalized Derivations

Javed et al. [1] introduced the notion of MA-semiring that is an additive inverse semiring satisfying A2 condition of Bandlet and Petrich [2]. ,e notion of MA-semirings is groundbreaking to use commutators and their related identities in semirings. ,is enables the algebraists to produce and extend some remarkable results in this area. MAsemiring 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]. ,e 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, onemay 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 S, we mean a semiring with absorbing “0,” in which addition is commutative. A semiring S is said to be additive inverse semring if for each s ∈ S there is a unique s′ ∈ S such that s + s′ + s � s and s′ + s + s′ � s′, where s′ denotes the pseudoinverse of s. An additive inverse semiring S is said to be an MA-semiring if it satisfies s + s′ ∈ Z(S), ∀s ∈ S, where Z(S) is the center of S. In fact, every ring is MA-semiring, while converse may not be true. ,e 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.


Introduction and Preliminaries
Javed et al. [1] introduced the notion of MA-semiring that is an additive inverse semiring satisfying A 2 condition of Bandlet and Petrich [2]. e notion of MA-semirings is groundbreaking to use commutators and their related identities in semirings. is enables the algebraists to produce and extend some remarkable results in this area. MAsemiring 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]. e concept of commutators along with derivations and certain additive mappings was further investigated and extended in [1,[3][4][5][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][10][11][12][13][14][15][16][17][18][19][20][21][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 S, we mean a semiring with absorbing "0," in which addition is commutative. A semiring S is said to be additive inverse semring if for each s ∈ S there is a unique s ′ ∈ S such that s + s ′ + s � s and s ′ + s + s ′ � s ′ , where s ′ denotes the pseudoinverse of s. An additive inverse semiring S is said to be an MA-semiring if it satisfies s + s ′ ∈ Z(S), ∀s ∈ S, where Z(S) is the center of S. In fact, every ring is MA-semiring, while converse may not be true. e 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.
{ } with addition ⊕ and multiplication ⊙ , respectively, defined by a ⊕ b � sup a, b { } and a ⊙ b � inf a, b { } is an MA-semiring. In fact, S is a commutative prime MA-semiring. roughout the paper, by S we mean an MA-semiring unless stated otherwise. We say S is prime if aSb � 0 implies that a � 0 or b � 0 and semiprime if aSa � 0 implies that a � 0. S is 2-torsion free if, for s ∈ S, 2s � 0 implies s � 0. An additive mapping * : S ⟶ S is involution if, ∀s, t ∈ S, (s * ) * � s and (st) * � t * s * . An element s ∈ S is Hermitian if s * � s and skew Hermitian if s * � s ′ . e set of Hermitian elements of S is denoted by H and that of skew Hermitian elements is denoted by K.
erefore, 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 d: . e concept of generalized derivation was studied for MA-semirings in [6]. An additive mapping [23]). e commutator is defined as [s, t] � st + t ′ s. By Jordan product, e following identities are very useful for sequel: for all s, t, z ∈ S, 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 S be a semiprime MA-semiring with involution * of second kind. en K ∩ Z(S) ≠ 0 and therefore H ∩ Z(S) ≠ 0. (1) For any k ∈ K, k 2 ∈ H (2) For any h ∈ H ∩ Z(S) and h o ∈ H, hh 0 ∈ H Idrissi and Oukhtite [25] proved some results on generalized derivations satisfying certain conditions on rings with involution. e main objective of this paper is to prove the results for MA-semirings with involution.

Main Results
Lemma 2. Let S be a 2-torsion free prime MA-semiring and let F d be a generalized derivation associated with a derivation then S is commutative.
Proof. By the hypothesis, for all s, t, r ∈ S, we have If d � 0, then (2) becomes In (3), replacing t by ts, we get From (3), we also have F d (s)t + F d (t)s ′ ∈ Z(S) and therefore (4) becomes In (5), replacing r by rw and using (5), we obtain By the primeness of S, we get [s, w] � 0 or it also implies In (7), replacing t by tr, we get F d (s)tr + F d (t)rx ′ � 0 and, using (8), we obtain In (9), replacing s by sw and using (9) again, we get erefore, S is commutative. We now consider the case when d ≠ 0. In (2), replacing t by ts and using (2) again, we have which further gives and, using (10), we obtain Using (11) into (14),

Lemma 3. Let S be a 2-torsion free prime MA-semiring with involution of second kind. If S satisfies
then S is commutative.
Proof. By the hypothesis, for all s, t, r ∈ S, we have In (16), replacing t by k ∈ K ∩ Z(S), we get and, using (17) then S is commutative.

Journal of Mathematics
Using (28) then S is commutative.
where I ′ (s) � s ′ � pseudo inverse of s. We now prove that ψ is generalized derivation. Firstly, for any s, t ∈ S, is shows that ψ is additive. Secondly, for any s, t ∈ S, is shows that ψ is generalized derivation associated with the derivation d. From (29), we can write ψ s, s * , r � 0, ∀s, r ∈ S. (33) Hence, by eorem 1, we conclude that S is commutative.

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

Proposition 2. Let
then S is commutative.
Proof. Linearizing (35) and using (35) again, we get And, replacing t by t * , we further get Suppose that d � 0. In (37), replacing t by erefore, in the view of Lemma 1, by the primeness of S, we obtain and this implies Using (39) into (37) and then using 2-torsion freeness of S, we obtain In (40) is proves that R is commutative.
Suppose that d ≠ 0. In (37), replacing t by yh, h ∈ H ∩ Z(S), we obtain and therefore Using (37) again, we obtain In (44), replacing t by tk, k ∈ K ∩ Z(S), we obtain In view of Lemma 1, by the primeness of S, we have and therefore From (49), we can write In (49), replacing t by tr, we get Using (50), we get By the primeness of S, we have that either S is commutative or d(h) � 0. Suppose that d(h) � 0. Following the same arguments as those in eorem 1, we have d(k) � 0, ∀k ∈ K ∩ Z(S). In (37), replacing t by tk, k ∈ K ∩ Z(S) and Using (53) into (37) and then using 2-torsion freeness of S, we get F d s°t , r � 0. (54) In (54), replacing t by h ∈ H ∩ Z(S) and using the fact that d(h) � 0, we obtain [F d (s), r] � 0 and hence, replacing s by [u, v], we find [F d [u, v], r] � 0. By Lemma 2, S is commutative.

Proposition 3. Let S be 2-torsion free prime MA-semiring with involution of second kind and let F d be a generalized derivation associated with a derivation d of S. If F d ≠ I and
then S is commutative.
Proof. Linearizing (58) and using (58) again, we get and hence, replacing t by t * , we further get In (60) then S is commutative.

Theorem 3. Let S be 2-torsion free prime MA-semiring with involution of second kind and let F d be a generalized derivation associated with a derivation d of S. If F d satisfies
then S is commutative.
Proof. When F d � 0, (67) becomes [[s, s * ], r] � 0, ∀s, r ∈ S. erefore, employing Lemma 4, we conclude that S is commutative.When F d ≠ 0, firstly suppose that d � 0. Linearizing (67) and using it again, we obtain And, replacing t by t * , we obtain In (69), replacing t by tk, k ∈ K ∩ Z(S) and using the assumption d � 0, we obtain By the primeness of S, we get As S is 2-torsion free, using (72) into (69), we get � 0 and therefore as above we conclude that either S is commutative or d(h) � 0 and this further implies d(k) � 0, ∀k ∈ K ∩ Z(S). In (69), replacing t by th, h ∈ H ∩ Z(S) and using the fact that In (69), replacing t by tk, k ∈ K ∩ Z(S) and using the fact Using (81) In (84), replacing t by t * , we obtain In (85), replacing t by tk, k ∈ K ∩ Z(S) and using the assumption d � 0, we get 6 Journal of Mathematics and therefore, by rearrangement, we have and hence In view of Lemma 1, by the primeness of S, we obtain which further gives Using (90) Equation (98) is the same as (79) of eorem 3; therefore, following the same steps, we conclude that either S is commutative or Rearranging the terms of (85), we get In (99), replacing t by tk, k ∈ K ∩ Z(S) and using the fact that d(k) � 0, we get In (104), replacing t by h ∈ H ∩ Z(S), we obtain 2h[s, r] � 0 and by the 2-torsion freeness of S, it is implied that h[s, r] � 0 and therefore hS[s, r] � 0. In view of Lemma 1, by the primeness of S, we conclude that S is commutative.
In view of the above results, we can easily conclude the following results.

Concluding Remarks
Commutativity is a very important aspect of mathematics and is discussed in almost all of its branches. is article presents some results on generalized derivations of MAsemirings with involution of second kind. is 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
e authors declare that they have no conflicts of interest.