Some results on commutativity of MA-semirings

Objective: The main aim of this article is to extend the concept of involution for a certain class of semirings known as MA-semirings. Now a days the commutativity conditions in the theory of rings and semirings becomes crucial for researchers. This motivates us to discuss some conditions onMA-semirings with involution which enforces commutativity. Method: We use the tools of derivations and involutions of second kind on MA-semirings. Findings: We are able to find the conditions of commutativity in semirings through these particular mappings. Novelty: To define the concept of Hermitian elements in MA-semirings with involution and to establish some commutativity results through different conditions involving Hermitian elements is the novel idea.


Introduction and Preliminaries
Involution plays a vital role in defining the important structures like B*-algebra and C*algebra. The rings with involution are also investigated by many algebraists (see (1)(2)(3)(4)(5) ). The concept of involution was also discussed for some other types inverse semirings (see (6)(7)(8)(9) ). The notion of MA-semiring was introduced by M.A Javed, M. Aslam and M. Hussain (10) . The theory of commutators along with derivations and certain additive mappings was further investigated in (10)(11)(12)(13)(14)(15) . For more on MA-semirings and MAsemirings with involution one can see (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) . Main objective of this paper is to introduce the notion of involution of First and Second kind for MA-semirings and generalize some results of Ring theory published in (25) . Now we include some necessary preliminaries for completion. By a semiring R, we mean a semiring with absorbing zero '0' in which addition is commutative. A semiring R is said to be additive inverse semiring if for each x ∈ R there is a unique x ′ ∈ R such that x+x ′ +x = x and x ′ +x+x ′ = x ′ and x ′ is called the Pseudo inverse of x . An additive inverse semiring R is said to be an MA-semiring if it satisfies x + x ′ ∈ Z(R), ∀x ∈ R , where Z(R) is the center of R.
Following example shows that every ring with absorbing zero '0' is an MA-semiring but converse may not be true in general. Example 1.1. (10) The set R = {0, 1, 2, 3, . . .}with addition ⊕ and multiplication ⊗ respectively defined by a ⊕ b = sup{a, b} and a ⊗ b = in f {a, b} is an MA-semiring which is not a ring. In fact Ris a commutative prime MA-semiring.
Such examples motivate us to generalize the results of ring theory in MA-semirings. Throughout the paper by a semiring Rwe mean a MA-semiring unless mentioned otherwise. Ris prime if aRb = {0} implies that a = 0 or b = 0 and semiprime if aRb = {0}implies that a = 0. Ris 2-torsion free if for x ∈ R, 2x = 0 implies x = 0. An additive mapping * : R → R is said to be involution if ∀x, y ∈ R, (x * ) * = x and (xy) * = y * x * . The set H = {x ∈ R : x * = x} is the set of all Hermitian elements and S = {x ∈ R : x * = x ′ } is the set of all Skew Hermitian elements. An additive mapping d : R → R is derivation if d(xy) = d(x)y + xd(y). We define commutator as [x, y] = xy + y ′ x. By Jordan product we mean all x, y, z ∈ R (For more see (10,14,15) ).

Main Results
First we introduce the notions of skew hermitian elements, involution of first and second kind for MA-semirings which are the generalization of the relevant concepts of rings.
then R is commutative. Proof. Linearizing (1), we get [x, y * ] + [y, x * ] = 0, ∀x, y ∈ R and replacing y by y * , we get In (2) replacing y by ys, Using (3) into (2) and hence by the 2−torsion freeness of R, we get [x, y] = 0, ∀x, y ∈ R which clearly shows that R is commutative. Lemma 2.5. Let Rbe a 2 − torsion free prime semiring with involution * of second kind and d be a nonzero derivation Linearizing (4) and using it again, we have Replacing y by y * in (5), we get Replacing https://www.indjst.org/ Replacing y by ys, s ∈ S ∩ Z − {0} in (6) and using it again, we get ( As (8) Using (9) into (7), we get 2[x, y]d(h) = 0, ∀x, y ∈ R and by the 2−torsion freeness of R, we have In (10), replacing y by yr and using (10) In (11) Since for each z ∈ Z, z + z * ∈ H ∩ Z , therefore from (11), we have Also since for each z ∈ Z, z ′ + z * ∈ H ∩ Z , therefore from (12), we have d(z ′ ) + d (z * ) = 0, ∀z ∈ Z , which further gives Using (14) into (13) and 2 − torsion freeness of R, we get In (6) From (6), we have Using (17) into (16), we get d[x, y] Replacing t by st in (21) and using it again, we get Replacing t by rt in (22) and using it again, we get [s, x] R[s,t] = {0}, ∀s,t, x ∈ R. By the primeness of R, we conclude that [s,t] = 0, ∀ s,t ∈ R and hence R is commutative. Theorem 2.6. Let R be a 2−torsion free prime semiring with involution ′ * ′ of second kind. If dis a derivation such that then R is commutative.

Concluding Remarks
The word commutativity carries much importance for every noncommutative algebraic structure. This article discusses some conditions on MA-semirings with involution especially of second kind which guarantee commutativity. The commutativity of algebraic structures brings convenience in calculations. Therefore this research is useful and opens the door of further research in this area by using different types of additive mappings. https://www.indjst.org/