Multiplicative Maps on Generalized n-matrix Rings

TL;DR

This paper investigates conditions under which multiplicative maps on generalized n-matrix rings are additive, and explores their properties as isomorphisms and derivations, expanding understanding of ring homomorphisms.

Contribution

It establishes a new condition ensuring multiplicative maps are additive on generalized n-matrix rings, and applies this to study m-multiplicative isomorphisms and derivations.

Findings

Multiplicative maps are additive under certain conditions.

Characterization of m-multiplicative isomorphisms on generalized n-matrix rings.

Analysis of m-multiplicative derivations in the same context.

Abstract

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.