A-Rings
Volume 96 / 2003
Colloquium Mathematicum 96 (2003), 277-292
MSC: 20K30, 20K20, 20K15.
DOI: 10.4064/cm96-2-10
Abstract
A ring $R$ is called an E-ring if every endomorphism of $R^{+},$ the additive group of $R,$ is multiplication on the left by an element of $R.$ This is a well known notion in the theory of abelian groups. We want to change the “E” as in endomorphisms to an “A” as in automorphisms: We define a ring to be an A-ring if every automorphism of $R^{+}$ is multiplication on the left by some element of $R.$ We show that many torsion-free finite rank (tffr) A-rings are actually E-rings. While we have an example of a mixed A-ring that is not an E-ring, it is still open if there are any tffr A-rings that are not E-rings. We will employ the Strong Black Box [5] to construct large integral domains that are A-rings but not E-rings.
