Additive Maps on Units of Rings

Let R be a ring. A map f: R -> R is additive if f (a + b) = f (a) + f (b) for all elements a and b of R. Here, a map f: R R is called unit-additive if f (u + v) = f (u) + f (v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of M-n, (F) is additive for all n >= 2, this paper is about the question of when every unit-additive map of a ring is additive. It is proved that every unit-additive map of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to Z(2) or R/J(R) congruent to Z(2) with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of Mn (R) is additive for all n >= 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f (uv) = f (u)f (v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.