On finite rings with a certain number of centralizers

Let Cent(R) denote the set of all distinct centralizers in a ring R. A ring R is said to be n-centralizer ring if |Cent(R)|=n, where n∈N. In this paper, we illustrate some connections between the centralizers and pairwise non-commuting elements of a finite ring. Besides that, we characterize all n-centralizer finite rings for n=6,7,8,9, and also compute the commuting probability for all n-centralizer finite rings for n=6,7,8,9.