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

<jats:p>Let R be a finite ring and let X be a non-empty subset of R. If ab=ba for any two distinct a,b∈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 suchsets. In this paper, we study the structures for some finite rings with maximal size of pairwise non-commuting elements is 5.</jats:p>