On a Class of Nowhere Commutative Semigroup Rings

For a finite algebraic structure A the commutativity degree of A, denoted by Pr(A), is the probability that two elements of A commute. Since 1973, Pr(A) studied for finite groups showing that Pr(A) ≤ 58 . The study of ”almost commutative” semigroups in 2011 showed that for some finite semigroup A, Pr(A) > 58 and even Pr(A) may be arbitrarily close to 1. Since 0 ≤ Pr(A) ≤ 1 then looking for algebraic structures A such that Pr(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 An,p = Zp(Pn) of order pn where, Pn is a non-commutative semigroup and show that Pr(An,p) → 0, for sufficiently large values of n and p. We name such semigroup rings as ”extremely non-commutative” semigroup rings.


Introduction
For a given finite algebraic structure A (group, ring, semigroup or semigroup ring), the commutativity degree of A is defined to be The study of this probability for finite groups started in 1973 by Gustafson [5] by showing that P r (A) ≤ 5  8 for every non-abelian group A. The commutativity for finite rings defined and studied by McHale in 1976 (see [8]).The article [7] studied the case when P r (A) ≥ 1  2 in 1995 for a finite group A. In the article [4] the case P r (A) < 1 2 and specially when 1 4 < P r (A) < 1 2 studied in 2008 by giving certain infinite classes of finite groups.One of the three studied classes in this article satisfies the condition P r (A) → 0 for the sufficiently large group A.
The study of this probability for semigroups started in 2011 by Ahmadidelir [1] where the authors proved that for some semigroup A, P r (A) > 5  8 .The natural question that "for which algebraic structure A (except the groups), P r (A) → 0" may be posed here and in what follows we construct a class of two parametric semigroup rings A n,p of order p n , for every integer n ≥ 2 and odd prime p, to show that P r (A n,p ) → 0. We name such an algebraic structure the "extremely non-commutative" algebraic structure.
Our notation is fairly standard and follows [2,3,6,9] on the presentations of groups and semigroups.We may also recall the definition of a group ring R[S] for a commutative ring R and an arbitrary finite semigroup S, consisting of all linear combinations g∈S α(g)g, where α(g) ∈ R and α(g) = 0 except for a finite number of coefficients.Of course the sum and multiplication of elements in R[S] are defined as usual by: Let n ≥ 2 be an integer and p an odd prime.Consider the presentation where, the indices are reduced modulo n.The semigroup defined by π (indeed, Sg(π), as a usual notation of the finitely presented structures), will be dented

Lemma 2.2 The semigroup ring A n,p contains exactly p n−1 elements satisfying n i=1 α i ≡ 0(mod p).
Proof There are exactly p n−1 sets as A(α 1 , α 2 , . . ., α n−1 ).On the other hand, there are a one-one correspondence between the elements of the set A n,p and the set So, the result follows at once by using the Lemma 2.1.

Lemma 2.3 Two different elements
Proof Considering the relators of the semigroup P n we get: Consequently, the last equation holds if and only if n i=1 α i ≡ 0(mod p) and n i=1 β i ≡ 0(mod p), for, x = y.

Results and Discussion
By using the results of Section 2 we give here the proofs of Proposition by P n .Finally, by denoting the semigroup ring Z p [P n ] by A n,p we give our main results as: The semigroup ring A n,p is finite of order p n .Moreover, for every integer n ≥ 2, the inequalities 1 2p 2 ≤ P r (A n,p ) ≤ 1 p 2 hold, for every odd prime p.The semigroup ring A n,p is extremely non-commutative.
The first part is obvious because of the representation of the elements of A n,p .For the second part we have to compute P then, for the sufficiently large values of p, P r (A n,p ) → 0 showing that A n,p is an extremely non-commutative semigroup ring.
r (A n,p ), for a given n and p.By the definition we get thatP r (A n,p ) = |A n,p | + |{(x, y) ∈ A 2 n,p |x = y, xy = yx}| |A 2 n,p | .Considering the Lemma 2.3 gives us the cardinal of the set {(x, y) ∈ A 2 n,p |x = y, xy = yx} as p n−1 (p n−1 −1)