Some results on commutativity of MA-semirings

Abstract

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 on MA-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.

A study on derivations of inverse semirings with involution

“…In this paper, we generalize some results of [4] and [15], for the case of Lie ideals of additively inverse semirings. Throughout this paper, S is a prime additively inverse semiring with A 2 − condition [2] i.e., for all a ∈ S, a + a 0 ∈ Z(S), where Z(S) is the center of S. Note that, an additively inverse semiring with A 2 − condition is also known as a MA-semiring.…”

“…The upcoming result is a generalization of[15, Lemma 2.4].Proposition 2.3. If S is with second kind involution and [a, a ] = 0, for all a ∈ L, then [L, S] = (0).…”

“…This section deals with the behaviour of derivations on Lie ideals of additively inverse semirings with involution. Also, it is discussed that how derivations effect the commutativity of additivity inverse semirings with involution and some results of [4], [15] are also generalized. In this section, L is a nonzero as well as 2-Lie ideal, S is a prime additively inverse semiring with involution and char(S) 6 = 2.…”

“…In 1998, Beidar and Martindale [5] examined some functional identities in prime rings with involution. In 2020, Ali et al proved some results concerning derivation and discussed certain differential identities of these semirings to analyse the commutativity of MA-semirings (see [4], [14], [15]). They assert an open question in [4] that is "How to control conditions of semirings which enable to induce the commutativity through Lie and other certain ideals of semirings?"…”

“…The set of hermitian elements of S is denoted by H and skew hermitian elements is denoted by S 1 . In addition, an involution is of second kind if Z(S)H. According to Ali et al [15], S 1 ∩ Z(S) 6 = (0) and H ∩ Z(S) 6 = (0) for semiprime additively inverse semiring with second kind involution and an ideal I of S is a -ideal, if I = I .…”