  Let $\mathfrak{R}$ and $\mathfrak{R}'$ be two associative rings (not
necessarily with the identity elements). A bijective map $\varphi$ of
$\mathfrak{R}$ onto $\mathfrak{R}'$ is called a \textit{$m$-multiplicative
isomorphism} if {$\varphi (x_{1} \cdots x_{m}) = \varphi(x_{1}) \cdots
\varphi(x_{m})$} for all $x_{1}, \cdots ,x_{m}\in \mathfrak{R}.$ In this
article, we establish a condition on generalized $n$-matrix rings, that assures
that multiplicative maps are additive on generalized $n$-matrix rings under
certain restrictions. And then, we apply our result for study of
$m$-multiplicative isomorphism and $m$-multiplicative derivation on generalized
$n$-matrix rings.
