Multiplicative Maps on Generalized n-matrix Rings

Let $\mathfrak{R}$ and ${\mathfrak{R}}^{\mathrm{\prime }}$ be two associative rings (not necessarily with the identity elements). A bijective map $\phi$ of $\mathfrak{R}$ onto ${\mathfrak{R}}^{\mathrm{\prime }}$ is called a \textit{$m$-multiplicative isomorphism} if {$\varphi (x_{1} \cdots x_{m}) = \varphi(x_{1}) \cdots \varphi(x_{m})} for allx_{1}, \cdots ,x_{m}\in \mathfrak{R}.$ 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.