Multiplicative Maps on Generalized n-matrix Rings

Abstract

Let $\mathfrak{R}$ and ${\mathfrak{R}}^{\prime }$ be two associative rings (not necessarily with the identity elements). A bijective map $\phi$ of $\mathfrak{R}$ onto ${\mathfrak{R}}^{\prime }$ is called a \textit{$m$-multiplicative isomorphism} if {$\phi (x_1\cdots x_m)=\phi (x_1)\cdots \phi (x_m)$} 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.

14 pages