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

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