ON COMMUTATIVITY OF SEMIRINGS

Abstract

We prove the following results. (1) If R is a semiring such that (ab) kk

k

=

a b 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 nnnnnn

a (n+1)! – torsion free semiring such that (ab) +b a =(ba) +a b is central for all a,b ∈ R then R is semicommutative. (3) If R is a

n! – torsion free semiring such that a b+b a = ba +ab for all a,b ∈ R nnn

or (ab) = a b for all a, b ∈ R then R is semicommutative.