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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-2-10,
author = {Manfred Dugas and Shalom Feigelstock},
title = {A-Rings},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {277-292},
zbl = {1037.16021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-2-10}
}
Manfred Dugas; Shalom Feigelstock. A-Rings. Colloquium Mathematicae, Tome 96 (2003) pp. 277-292. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm96-2-10/