Some results on the finite rings with maximal size of pairwise non-commuting elements is 5

Abstract

Let \(R\) be a finite ring and let \(X\) be a non-empty subset of \(R\). If \(ab\neq ba\) for any two distinct \(a,b\in X\), then \(X\) is called a set of pairwise non-commuting elements of \(R\). Moreover, \(X\) is said to be a set of pairwise non-commuting elements of \(R\) with maximal size if its cardinality is the largest one among all such sets. In this paper, we study the structures for some finite rings with maximal size of pairwise non-commuting elements is \(5\).