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

Let \(R\) and \(R'\) be two associative rings (not necessarily with identity elements). A bijective map \(φ\) of \(R\) onto \(R'\) is called an \(m\)-multiplicative isomorphism if {\(φ (x₁ ⋯ xₘ) = φ(x₁) ⋯ φ(xₘ)\)} for all \(x₁, ,xₘ∈ R.\) 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.