  Let $S, K$ be two subrings of a finite ring $R$. Then the generalized
non-commuting graph of subrings $S, K$ of $R$, denoted by $\Gamma_{S, K}$, is a
simple graph whose vertex set is $(S \cup K) \setminus (C_K(S) \cup C_S(K))$
and two distinct vertices $a, b$ are adjacent if and only if $a \in S$ or $b
\in S$ and $ab \neq ba$. We determine the diameter, girth and some dominating
sets for $\Gamma_{S, K}$. Some connections between the $\Gamma_{S, K}$ and
$\Pr(S, K)$ are also obtained. Further, $\Z$-isoclinism between two pairs of
finite rings is defined and showed that the generalized non-commuting graphs of
two $\Z$-isoclinic pairs are isomorphic under some condition.
