On finite rings with a certain number of centralizers

Abstract

Let [Formula: see text] denote the set of all distinct centralizers in a ring [Formula: see text]. A ring [Formula: see text] is said to be [Formula: see text]-centralizer ring if [Formula: see text], where [Formula: see text]. In this paper, we illustrate some connections between the centralizers and pairwise non-commuting elements of a finite ring. Besides that, we characterize all [Formula: see text]-centralizer finite rings for [Formula: see text], and also compute the commuting probability for all [Formula: see text]-centralizer finite rings for [Formula: see text].