| Abstract: | Abstract Let R be a ring with identity such that R +, the additive group of R, is torsion-free of finite rank (tffr). The ring R is called an E-ring if End(R +) = {x ? ax : a ∈ R} and is called an A-ring if Aut(R +) = {x ? ux : u ∈ U(R)}, where U(R) is the group of units of R. While E-rings have been studied for decades, the notion of A-rings was introduced only recently. We now introduce a weaker notion. The ring R, 1 ∈ R, is called an AA-ring if for each α ∈ Aut(R +) there is some natural number n such that α n ∈ {x ? ux : u ∈ U(R)}. We will find all tffr AA-rings with nilradical N(R) ≠ {0} and show that all tffr AA-rings with N(R) = {0} are actually E-rings. As a consequence of our results on AA-rings, we are able to prove that all tffr A-rings are indeed E-rings. |
