On commuting probability of finite rings

The commuting probability of a finite ring R, denoted by Pr(R), is the probability that any two randomly chosen elements of R commute. In this paper, we obtain several bounds for Pr(R) through a generalization of Pr(R). Further, we define Z-isoclinism between two pairs of rings and show that the generalized commuting probability, defined in this paper, is invariant under Z-isoclinism between two pairs of finite rings.