On Jordan Ideals of Inverse Semirings with Involution

Abstract

Objectives:The main objective of this article is to introduce *-Jordan ideals of a certain class of semirings called MA-semirings with involution and to investigate some conditions for which the above said ideals contained in the center. Methods and findings: We use the Jacobian identities and 2-torsion freeness of MA semirings. In this connection, we establish some important results of ring theory for the class of MA-semirings. Applications/ improvements: The commutative property is helpful to study the theory of semirings with ease therefore we find some conditions to impose commutativity in semirings, which are indeed novel idea in the field of semirings. Furthermore, these conditions are used in a most generalized way that these conditions bring the *-Jordan ideals to the center, therefore, it would be the corollary of result that semiring is commutative.

(α, β) – Derivations on Prime Inverse Semirings

“…If where r, s then Proof If r = s , where r, s , then by adding (s + s') to the two sides, we get r , for all Again, since α is an automorphism on S, we get for all therefore on S. Lemma 2.5. [17] Let J be a nonzero Jordan ideal of S, then and Lemma 2.6.…”

Some results on commutativity of MA-semirings

“…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) .…”