AA-Rings

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.

Key Words:

• E-Rings
• A-Rings
• AA-Rings

1991 Mathematics Subject Classification:

• 20K30
• 20K15

Acknowledgments

Research partially supported by Baylor University's Summer Sabbatical Program. Special thanks to Anja Möhring for her translation of Figer (1986), which provided an essential clue for this paper.

Notes

^#Communicated by K. Rangaswamy.