On a class of nowhere commutative semigroup rings

Abstract

For a finite algebraic structure A the commutativity degree of A, denoted by P r (A), is the probability that two elements of A commute. Since 1973, P r (A) studied for finite groups showing that P r (A) ≤ 5 8 . The study of "almost commutative" semigroups in 2011 showed that for some finite semigroup A, P r (A) > 5 8 and even P r (A) may be arbitrarily close to 1. Since 0 ≤ P r (A) ≤ 1 then looking for algebraic structures A such that P r (A) → 0, is of interest, for, the centralizer of every element of such A should be a singleton. In this paper for every integer n ≥ 2 and every prime p ≥ 3 we give an infinite class of finite semigroup rings A n,p = Z p (P n ) of order p n where, P n is a non-commutative semigroup and show that P r (A n,p ) → 0, for sufficiently large values of n and p. We name such semigroup rings as "extremely non-commutative" semigroup rings.