Multiplicative maps on generalized \(n\)-matrix rings

Abstract

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