Central endomorphisms of groups and radical rings

An endomorphism \gamma of a group G is called a central endomorphism if x^{-1}x^{\gamma} lies into the centre Z(G) of G for each element x of G. It is easy to show that all non-zero central endomorphisms of G are automorphisms if and only if the ring R=Hom(G, Z(G)) of all homomorphisms of G into Z(G) is a (Jacobson) radical ring, i.e. R is a group via the circle operation f \circ g = f + g + fg for all f, g \in R. In this article it is proved that R is an artinian radical ring if and only if G has no non-trivial abelian direct factors and the additive group of R satisfies the minimal condition on subgroups. Moreover, it is pointed out how central automorphisms can be used to produce solutions of the Yang-Baxter equation.