On Generalized (α, β) Derivation on Prime Semirings

Abstract

In this paper we introduce generalized (α, β) derivation on Semirings and extend some results of Oznur Golbasi on prime Semiring. Also, we present some results of commutativity of prime Semiring with these derivation.

Dependent Element and Free Actions of Centralizer and Reverse Centralizer on Prime and Semiprime Semirings

“…Replace u by u w, where w ∈ I, Since S is cancellative we have, for all v, w ∈ I. Then I is commutative, by Lemma 2.22 [16] we get S is commutative. Similarly proved when T is right centralizer.…”

(α, β) – Derivations on Prime Inverse Semirings

“…After that, many researchers studied classes of semiring, including Golan [3] and Fang [4]. Recently, many authors studied such semirings in various ways, describing the analysis of prime and semiprime semirings with various types of derivations [5][6][7][8][9][10].…”

“…Recall that S is called a prime inverse semiring if , for all implies that either or It is also called semiprime inverse semiring if , for all implies that , or S has no non-zero nilpotetent ideal [9]. We call S as 2-tortion free if implies that An additive mapping is called (α, β)derivation" of S if for all r, s S, where α, β are automorphisms on S [8].…”