Multiplicative mappings of rings

Let R and S be arbitrary associative rings. A mapping phi of R onto Y is called a multiplicative isomorphism if. is bijective and satisfies phi(xy) = phi(x)phi(y) for all x, y epsilon 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 phi 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.