Multiplicative Mappings of Rings

Abstract

Let ℛ and ${\fancyscript S}$ be arbitrary associative rings. A mapping φ of ℛ onto ${\fancyscript S}$ is called a multiplicative isomorphism if φ is bijective and satisfies φ(xy) = φ(x)φ(y) for all x, y ∈ ℛ. In this short note, we establish a condition on ℛ, in the case where ℛ may not contain any non–zero idempotents, that assures that φ is additive, which generalizes the famous Martindale's result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.