# Commutativity of MA-Semirings with Involution through Generalized Derivations

Authors: Liaqat Ali; Muhammad Aslam; Muhammad Imran Qureshi; Yaqoub Ahmed Khan; Shafiq Ur Rehman; Ghulam Farid
Journal: Journal of Mathematics
 (2020)

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## 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.


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## Body

## 1. Introduction and Preliminaries

Javed et al. [1] introduced the notion of MA-semiring that is an additive inverse semiring satisfying A2 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 semiringS, 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′∈ZS,∀s∈S, where ZS is the center of S. 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 S=0,1,2,3,4,…. with addition ⊕ and multiplication ⊙, respectively, defined by a⊕b=supa,b and a⊙b=infa,b is an MA-semiring. In fact, S is a commutative prime MA-semiring.
Throughout the paper, byS 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′. The set of Hermitian elements of S is denoted by H and that of skew Hermitian elements is denoted by K.Example 2.
(see [5]). Let S,+,⋅ be an MA-semiring. Then the set S with addition “+” and multiplication ⋅ defined as a⋅b=b.a forms an MA-semiring called the opposite MA-semiring of S. We usually denote it as S°. Consider R=S×S∘ with a,b⊕c,d=a+c,b+d and a,b⊙c,d=a.c,b⋅d=ac,db. Then R,⊕,⊙ forms an MA-semiring. Define ∗:R⟶R by s,t∗=t,s. Then ∗ defines an involution on R. Thus, the triplet R,⊕,⊙ 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 mappingd:S⟶S is a derivation if dst=dst+sdt. The concept of generalized derivation was studied for MA-semirings in [6]. An additive mapping Fd:S⟶S is a generalized derivation associated with a derivation d if Fdst=Fdst+sdt (see [23]). The commutator is defined as s,t=st+t′s. By Jordan product, we mean s°t=st+ts for all s,t∈S. A mapping f:S⟶S is commuting if fs,s=0, ∀s∈S. A mapping f:S⟶S is centralizing if fs,s,r=0, ∀s,r∈S. The following identities are very useful for sequel: for all s,t,z∈S,(1)
s,st=ss,t(2)
st,z=st,z+s,zt(3)
s,tz=s,tz+ts,z(4)
s,t+t,s=ts+s′=st+t′(5)
st′=s′t=st′(6)
s,t′=s,t′=s′,t(7)
s∘t+z=s∘t+s∘z
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]).
LetS be a semiprime MA-semiring with involution ∗ of second kind. Then K∩ZS≠0 and therefore H∩ZS≠0.Remark 1.
LetS be an MA-semiring with involution ∗ of second kind.(1)
For anyk∈K, k2∈H(2)
For anyh∈H∩ZS and ho∈H, hh0∈H
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.
LetS be a 2-torsion free prime MA-semiring and let Fd be a generalized derivation associated with a derivation d of S. If Fd satisfies(1)Fds,t,r=0,∀s,t,r∈S,then S is commutative.Proof.
By the hypothesis, for alls,t,r∈S, we have Fdst+Fdts′,r=0 and therefore(2)Fdst+sdt+Fdts′+tds′,r=0.
Ifd=0, then (2) becomes(3)Fdst+Fdts′,r=0.
In (3), replacing t by ts, we get(4)Fdst+Fdts′s,r=0.
From (3), we also have Fdst+Fdts′∈ZS and therefore (4) becomes(5)Fdst+Fdts′s,r=0.
In (5), replacing r by rw and using (5), we obtain(6)Fdst+Fdts′Ss,w=0.
By the primeness ofS, we get s,w=0 or Fdst+Fdts′=0. If s,w=0, then S is commutative. Secondly, if(7)Fdst+Fdts′=0,it also implies(8)Fdst=Fdts.
In (7), replacing t by tr, we get Fdstr+Fdtrx′=0 and, using (8), we obtain(9)Fdts,r=0.
In (9), replacing s by sw and using (9) again, we get FdtSw,r=0. As S is prime and Fd≠0, w,r=0. Therefore, S is commutative.
We now consider the case whend≠0. In (2), replacing t by ts and using (2) again, we have(10)Fds,ts,r+s,t,rds=0,which further gives(11)Fds,ts,r=s,t,r′ds.
In (10), replacing r by rw, we obtain(12)Fds,trs,w+Fds,ts,rw+rs,t,wds+s,t,rwds=0.
From (2), using Fds,t∈ZS in the last expression, we get(13)rFds,ts,w+s,t,wds+Fds,ts,rw+s,t,rwds=0,and, using (10), we obtain(14)Fds,ts,rw+s,t,rwds=0.
Using (11) into (14), we get s,t,rw,ds=0 and so s,t,rSw,ds=0. By the primeness of S, we have either s,t,r=0 or w,ds=0. If w,ds=0, then dw,w=0,∀w∈G, and therefore, by Theorem 2.2 of [26], S is commutative. Secondly, if s,t,r=0, then, replacing t by ts, we obtain s,ts,r=0 and hence s,tSs,r=0. By the primeness of S, we conclude that S is commutative.Lemma 3.
LetS be a 2-torsion free prime MA-semiring with involution of second kind. If S satisfies(15)s∘t+t∗∘s∗,r=0,∀s,t,r∈S,then S is commutative.Proof.
By the hypothesis, for alls,t,r∈S, we have(16)s∘t+t∗∘s∗,r=0.In (16), replacing t by k∈K∩ZS, we get s∘k+k∗∘s∗,r=0, which implies s∘k+k′∘s∗,r=0 and therefore 2ks+s∗′,r=0. By the 2-torsion freeness of S, we have ks+s∗′,r=0, which further means kSs+s∗′,r=0. In view of Lemma 1, K∩ZS≠0. Therefore, by the primeness of S, we obtain s+s∗′,r=0, which is further simplified to be(17)s,r=s∗,r.
In (16), replacing t by h∈H∩ZS, we get s∘h+h∗∘s∗,r=0 and therefore 2hs+s∗,r=0. By the 2-torsion freeness of S, we get hs+s∗,r=0 and therefore hSs+s∗,r=0. In the view of Lemma 1, H∩ZS≠0; therefore, by the primeness of S, we have s+s∗,r=0 and hence(18)s,r+s∗,r=0,and, using (17) into (18), we obtain 2s,r=0 which, by the 2-torsion freeness of S, gives s,r=0. This proves that S is commutative.Theorem 1.
LetS be a 2-torsion free prime MA-semiring with involution ∗ and let Fd be a nonzero generalized derivation associated with a derivation d of S. If Fd satisfies(19)Fds,s∗,r=0,∀s,r∈S,then S is commutative.Proof.
Firstly, suppose thatd≠0. Linearizing (19) and using (19) again, we get(20)Fds,t∗+Fdt,s∗,r=0.
In (20), replacing t by t∗, we get(21)Fds,t+Fdt∗,s∗,r=0.
In (21), replacing t by tk,k∈K∩ZS, we get Fds,tk+Fdt∗,s∗k′,r=0. As d=0, kFds,t+Fdt∗,s∗′,r=0, which further implies kSFds,t+Fdt∗,s∗′,r=0. In view of Lemma 1, by the primeness of S, we get Fds,t+Fdt∗,s∗′r=0 and therefore(22)Fds,t,r=Fdt∗,s∗,r.
Using (22) into (21) and hence using 2-torsion freeness of S, we get Fds,t,r=0. and by Lemma 2 we conclude that S is commutative.
Secondly, suppose thatd≠0.
In (21), replacing t by yh,h∈H∩ZS, we obtain Fds,th+Fdt∗,s∗h,r=0 and so Fds,th+s,tdh+Fdt∗,s∗h+t∗,s∗dh,r=0. Rearranging the terms, we get(23)Fds,t+Fdt∗,s∗,rh+s,t+t∗,s∗dh,r=0.
Using (21) into (23), we get(24)s,t+t∗,s∗dh,r=0.
In (24), replacing t by tk,k∈K∩ZS, we obtain s,t+t∗,s∗′dh,rk=0 and therefore s,t+t∗,s∗′dh,rSk=0. By the primeness S, we have s,t+t∗,s∗′dh,r=0 and therefore(25)s,tdh,r=t∗,s∗dh,r.
Using (23) into (24) and hence using 2-torsion freeness of S, we get s,tdh,r=0, which can be further written as(26)s,t,rdh+s,tdh,r=0.
In (26), replacing t by st, we get ss,t,rdh+ss,tdh,r=0 and therefore ss,t,rdh+s,ts,rdh+ss,tdh,r=0. Rearranging the terms, ss,t,rdh+s,tdh,r+s,ts,rdh=0. Using (26) again, we obtain s,ts,rdh=0. Therefore, we can easily obtain s,ts,rSdh=0 which, by the primeness, gives that either S is commutative or dh=0. Suppose that(27)dh=0.
In the view of Lemma1, for any k∈K∩ZS, k2∈H∩ZS; therefore, replacing h by k2 in (27), we get kSdk=0, and, by the primeness of S, we have dk=0. Replacing t by tk,k∈K∩ZS in (21), we get Fds,t+Fdt∗,s∗′,rk=0 and therefore Fds,t+Fdt∗,s∗′,rSk=0. In the view of Lemma 1, using the primeness of S, we have Fds,t+Fdt∗,s∗′,r=0, which further implies(28)Fds,t,r=Fdt∗,s∗,r.
Using (28) into (21) and then using 2-torsion freeness of S, we obtain Fds,t,r=0. Employing Lemma 2, we get the required result.Proposition 1.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd≠I and(29)Fds,s∗+s,s∗′,r=0,∀s,r∈S,then S is commutative.Proof.
Suppose thatFd≠I. Define a mapping ψ:S⟶S by(30)ψs=Fds+I′s=Fds+s′,where I′s=s′=pseudo inverse ofs. We now prove that ψ is generalized derivation.
Firstly, for anys,t∈S,(31)ψs+t=Fds+t+s+t′=Fds+Fdt+s′+t′=Fds+s′+Fdt+t′=ψs+ψt.
This shows thatψ is additive. Secondly, for any s,t∈S,(32)ψst=Fdst+st′=Fdst+sdt+s′t=Fdst+s′t+sdt=Fds+s′t+sdt=ψst+sdt.
This shows thatψ is generalized derivation associated with the derivation d. From (29), we can write(33)ψs,s∗,r=0,∀s,r∈S.
Hence, by Theorem1, we conclude that S is commutative.On the similar lines of Proposition1, we can obtain the following proposition.Proposition 2.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd≠I′ and(34)Fds,s∗+s,s∗,r=0,∀s,r∈S,then S is commutative.Theorem 2.
LetS be a 2-torsion free prime MA-semiring with involution ∗. If Fd is a nonzero generalized derivation satisfying(35)Fds∘s∗,r=0,∀s,r∈S,then S is commutative.Proof.
Linearizing (35) and using (35) again, we get(36)Fds∘t∗+Fdt∘s∗,r=0,And, replacing t by t∗, we further get(37)Fds∘t+Fdt∗∘s∗,r=0.
Suppose thatd=0. In (37), replacing t by tk,k∈K∩ZS, we get Fds°tk+Fdt∗°s∗′k,r=0, which further gives Fds°t+Fdt∗°s∗′,rSk=0. Therefore, in the view of Lemma 1, by the primeness of S, we obtain(38)Fds∘t+Fdt∗∘s∗′,r=0,and this implies(39)Fds∘t,r=Fdt∗∘s∗,r.
Using (39) into (37) and then using 2-torsion freeness of S, we obtain(40)Fds∘t,r=0.
In (40), replacing t by h∈H∩ZS, we obtain Fds,rh=0 and therefore Fds,rSh=0. In view of Lemma 1, by the primeness, we obtain(41)Fds,r=0.
In (41), replacing s by st and using (41) again, we find Fdst,r=0 and therefore FdsSt,r=0. As Fd≠0, by the primeness, we have t,r=0. This proves that R is commutative.
Suppose thatd≠0.
In (37), replacing t by yh,h∈H∩ZS, we obtain(42)Fds∘t+t∗∘s∗h,r=0,and therefore(43)Fds∘t+t∗∘s∗,rh+s∘t+t∗∘s∗dh,r=0.
Using (37) again, we obtain(44)s∘t+t∗∘s∗dh,r=0.
In (44), replacing t by tk,k∈K∩ZS, we obtain(45)s∘t+t∗∘s∗′dh,rk=0.
In view of Lemma1, by the primeness of S, we have(46)s∘t+t∗∘s∗′dh,r=0,and therefore(47)s∘tdh,r=t∗∘s∗dh,r.
Using (47) into (44) and the 2-torsion freeness of S, we obtain(48)s∘tdh,r=0.
In (48), replacing t by st and using (48) again, we obtain s,rs∘tdh=0 and therefore s,rSs∘tdh=0, which further, by the primeness, implies that either R is commutative or(49)s∘tdh=0.
From (49), we can write(50)stdh=ts′dh.
In (49), replacing t by tr, we get(51)strdh+srtdh=0.
Using (50), we get(52)s,tSdh=0.
By the primeness ofS, we have that either S is commutative or dh=0. Suppose that dh=0. Following the same arguments as those in Theorem 1, we have dk=0, ∀k∈K∩ZS. In (37), replacing t by tk,k∈K∩ZS and using dk=0, we obtain Fds°t+Fdt∗°s∗′,r=0, which further implies(53)Fds∘t,r=Fdt∗∘s∗,r.
Using (53) into (37) and then using 2-torsion freeness of S, we get(54)Fds°t,r=0.
In (54), replacing t by h∈H∩ZS and using the fact that dh=0, we obtain Fds,r=0 and hence, replacing s by u,v, we find Fdu,v,r=0. By Lemma 2, S is commutative.Proposition 3.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd≠I and(55)Fds∘s∗+s∘s∗′,r=0,∀s,r∈S,then S is commutative.Proof.
Defineψ:S⟶S by ψs=Fds+I′s=Fds+s′, where I′s=s′. Then, following the same lines of the proof of Proposition 1, we find that ψ is a generalized derivation satisfying(56)ψs∘s∗,r=0.
Hence, by Theorem2, S is commutative.On the similar lines, we can obtain the following result.Proposition 4.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd≠I′ and(57)Fds°s∗+s°s∗,r=0,∀s,r∈S,then S is commutative.Lemma 4.
LetS be 2-torsion free prime MA-semiring with involution of second kind. If(58)s,s∗,r=0,∀s,r∈S,then S is commutative.Proof.
Linearizing (58) and using (58) again, we get(59)s,t∗+t,s∗,r=0,∀s,t,r∈S,and hence, replacing t by t∗, we further get(60)s,t+t∗,s∗,r=0.
In (60), replacing t by tk,k∈K∩ZS, we obtain s,t+t∗,s∗′,rk=0 and therefore s,t+t∗,s∗′,rSk=0. In view of Lemma 1, using the primeness of S, we have s,t+t∗,s∗′,r=0 and hence(61)s,t,r=t∗,s∗,r.
Using (61) into (60), we obtain 2s,t,r=0 and since S is 2-torsion free, (62)s,t,r=0.
Replacingt by st in (62) and using (62) again, we obtain s,tt,r=0 and therefore s,tSt,r=0, which, by the primeness, implies S is commutative.Lemma 5.
LetS be 2-torsion free prime MA-semiring with involution of second kind. If(63)s∘s∗,r=0,∀s,r∈S,then S is commutative.Proof.
Linearizing (63) and using (63) again, we obtain s°t∗+t°s∗,r=0 and hence, replacing t by t∗, we obtain(64)s∘t+t∗∘s∗,r=0.
In (64), replacing t by tk,k∈K∩ZS, we obtain s∘t+t∗∘s∗′,rk=0 and therefore s∘t+t∗∘s∗′,rSk=0. In view of Lemma 1, we get s∘t+t∗∘s∗′,r=0 and hence(65)s∘t,r=t∗∘s∗,r.
AsS is 2-torsion free, using (65) into (64), we get(66)s∘t,r=0.
In (66), replacing t by h∈H∩ZS, we get s,rSh=0 and, employing Lemma 1, we conclude that S is commutative.Theorem 3.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd satisfies(67)Fds∘s∗+s,s∗,r=0,∀s,r∈S,then S is commutative.Proof.
WhenFd=0, (67) becomes s,s∗,r=0,∀s,r∈S. Therefore, employing Lemma 4, we conclude that S is commutative.When Fd≠0, firstly suppose that d=0. Linearizing (67) and using it again, we obtain(68)Fds∘t∗+Fdt∘s∗+s,t∗+t,s∗,r=0,
And, replacingt by t∗, we obtain(69)Fds∘t+Fdt∗∘s∗+s,t+t∗,s∗,r=0.
In (69), replacing t by tk,k∈K∩ZS and using the assumption d=0, we obtain(70)Fds∘t+Fdt∗∘s∗′+s,t+t∗,s∗′,rSk=0.
By the primeness ofS, we get(71)Fds∘t+Fdt∗∘s∗′+s,t+t∗,s∗′,r=0,and hence(72)Fds∘t+s,t,r=Fdt∗∘s∗+t∗,s∗,r.
AsS is 2-torsion free, using (72) into (69), we get(73)Fds∘t+s,t,r=0.
From (73), we can write(74)Fds∘t,r=s,t,r′=t,s,r.
Interchangings and t in (74), we obtain Fdt∘s,r=s,t,r. But t∘s=s∘t; therefore(75)Fds∘t,r=s,t,r.
Using (75) into (74), we get 2s,t,r=0 and, by the 2-torsion freeness of S, we get(76)s,t,r=0.
In (76), replacing t by st and using (76) again, we obtain s,ts,r=0 and therefore s,tRs,r=0. By the primeness of S, we conclude that S is commutative.
Next, suppose thatd≠0.
In (69), replacing t by th,h∈H∩ZS and using (69) again, we obtain(77)s,t+t∗,s∗dh,r=0.
In (77), replacing t by tk,k∈K∩ZS, we obtain s,t+t∗,s∗′dh,rw=0 and therefore s,t+t∗,s∗′dh,rSw=0. In view of Lemma 1, by the primeness of S, we obtain s,t+t∗,s∗′dh,r=0, which further gives(78)s,tdh,r=t∗,s∗dh,r.
AsS is 2-torsion free, using (78) into (77), we get(79)s,tdh,r=0.
In (79), replacing t by ts and using (79), we obtain s,tdhs,r=0 and therefore s,tdhSs,r=0. By the primeness of S, we get s,tdh=0 and therefore as above we conclude that either S is commutative or dh=0 and this further implies dk=0,∀k∈K∩ZS. In (69), replacing t by th,h∈H∩ZS and using the fact that dh=0, we obtain Fds∘t+Fdt∗∘s∗,rh=0, which further implies Fds∘t+Fdt∗∘s∗,rSh=0. In view of Lemma 1, we have(80)Fds∘t+Fdt∗∘s∗,r=0.
In (69), replacing t by tk,k∈K∩ZS and using the fact that dk=0, we obtain Fds∘t+Fdt∗∘s∗′,rk=0, which further implies Fds∘t+Fdt∗∘s∗′,rSk=0. In view of Lemma 1, we obtain Fds°t+Fdt∗°s∗′,r=0 and hence(81)Fds∘t,r+Fdt∗∘s∗,r.
Using (81) into (80) and then by the 2-torsion freeness of S, we obtain Fds∘t,r=0. Following the same steps as Theorem 2 (equation (40)), we conclude that S is commutative.On the similar lines of Theorem3, we can establish the following result.Theorem 4.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd satisfies(82)Fds∘s∗+s,s∗′,r=0,∀s,r∈S,then S is commutative.Theorem 5.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. If Fd satisfies(83)Fds,s∗+s∘s∗,r=0,∀s,r∈S,then S is commutative.Proof.
Suppose thatFd=0. Then (83) becomes s°s∗,r=0,∀s,r∈S. By Lemma 5, S is commutative.
Now suppose thatFd≠0. Consider the case when d=0.
Linearizing (83) and using (83) again, we get(84)Fds,t∗+Fdt,s∗+s∘t∗+t∘s∗,r=0.
In (84), replacing t by t∗, we obtain(85)Fds,t+Fdt∗,s∗+s∘t+t∗∘s∗,r=0.
In (85), replacing t by tk,k∈K∩ZS and using the assumption d=0, we get(86)Fds,tk+Fdt∗,s∗k′+s∘tk+t∗∘s∗k′,r=0,and therefore, by rearrangement, we have(87)Fds,t+s∘t+Fdt∗,s∗+t∗∘s∗′,rk=0,and hence(88)Fds,t+s∘t+Fdt∗,s∗+t∗∘s∗′,rSk=0.
In view of Lemma1, by the primeness of S, we obtain(89)Fds,t+s∘t+Fdt∗,s∗+t∗∘s∗′,r=0,which further gives(90)Fds,t+s∘t,r=Fdt∗,s∗+t∗∘s∗,r,
Using (90) into (85) and the 2-torsion freeness of S, we obtain(91)Fds,t+s∘t,r=0.
In (85), interchanging s and t, we get Fdt,s+t∘s,r=0. But since s°t=t°s, Fdt,s+s∘t,r=0, which implies Fds,t′+s°t,r=0. Hence,(92)Fds,t,r=s∘t,r.
AsS is 2-torsion free, using (92) into (91), we find(93)s∘t,r=0.
In (93), replacing t by h∈H∩ZS and then using 2-torsion freeness of S, we get hs,r=0 and therefore hSs,r=0. In view of Lemma 1, by the primeness of S, we get s,r=0. This shows that R is commutative.
Next, suppose thatd≠0.
In (85), replacing t by yh,h∈H∩ZS, we get(94)Fds,th+s,tdh+Fdt∗,s∗h+t∗,s∗dh+s∘th+t∗∘s∗h,r=0.
Rearranging terms, we can write(95)Fds,t+Fdt∗,s∗+s°t+t∗°s∗,rh+s,t+t∗,s∗dh,r=0.
Using (85) again, we get(96)s,t+t∗,s∗dh,r=0.
In (96), replacing t by tk,k∈K∩ZS, we obtain s,t+t∗,s∗′dh,rk=0, which further gives s,t+t∗,s∗′dh,rSw=0. As S is prime, employing Lemma 1, we get s,tdh+t∗,s∗′dh,r=0 and hence(97)s,tdh,r=t∗,s∗dh,r.
Using (97) into (96) and hence using 2-torsion freeness of S, we obtain(98)s,tdh,r=0.
Equation (98) is the same as (79) of Theorem 3; therefore, following the same steps, we conclude that either S is commutative or dh=0. If dh=0, then dk=0,∀k∈K∩ZS.
Rearranging the terms of (85), we get(99)Fds,t+s∘t+Fdt∗,s∗+t∗∘s∗,r=0.
In (99), replacing t by tk,k∈K∩ZS and using the fact that dk=0, we get(100)Fds,t+s∘t+Fdt∗,s∗′+t∗∘s∗′,rk=0,which further implies(101)Fds,t+s∘t,r=Fdt∗,s∗+t∗∘s∗,r.
Using (101) into (99) and then by the 2-torsion freeness of S, we have(102)Fds,t+s∘t,r=0.
Interchangings and t in (102), we obtain Fdt,s+t∘s,r=0. As t∘s=s∘t, the last equation becomes Fds,t′+s∘t,r=0. This implies that(103)Fds,t=s∘t,r.
Using (103) into (102), we get 2s∘t,r=0 and by the 2-torsion freeness of S, we have(104)s∘t,r=0.
In (104), replacing t by h∈H∩ZS, we obtain 2hs,r=0 and by the 2-torsion freeness of S, it is implied that hs,r=0 and therefore hSs,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.Theorem 6.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. Then the following statements are equivalent:(i)
S,S=0(ii)
Fds∘s∗+s,s∗,r=0,∀s,r∈S(iii)
Fds,s∗+s∘s∗,r=0,∀s,r∈STheorem 7.
LetS be 2-torsion free prime MA-semiring with involution of second kind and let Fd be a generalized derivation associated with a derivation d of S. Then the following statements are equivalent:(i)
S,S=0(ii)
Fds°s∗+s,s∗′,r=0,∀s,r∈S(iii)
Fds,s∗+s∘s∗′,r=0,∀s,r∈S


## 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.



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