  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.
