ON COMMUTATIVITY OF SEMIRINGS

Abstract

We prove the following results. (1) If R is a semiring such that (ab)k = akbk for all a,b ∈ R and (i) fixed non negative integers k = n, n+1,n+2 or (ii) fixed positive integers k = m, m + 1,n, n + 1 where (m,n)=1 then R is semicommutative. If R is also additively cancellative then R is commutative. Thus we generalize the results of [7] and [2]. (2) If R is a( n + 1)! – torsion free semiring such that (ab)n + bnan =(ba)n + anbn is central for all a,b ∈ R then R is semicommutative. (3) If R is a n! – torsion free semiring such that anb+bna = ban +abn for all a,b ∈ R or (ab)n = anbn for all a,b ∈ R then R is semicommutative