A-Rings

Manfred Dugas¹ ; Shalom Feigelstock²

¹ Department of Mathematics Baylor University Waco, Texas 76798, U.S.A.
² Department of Mathematics Bar-Ilan University Ramat Gan, Israel

Colloquium Mathematicum, Tome 96 (2003) no. 2, pp. 277-292

Résumé

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.

Détail

DOI : 10.4064/cm96-2-10

Keywords: ring called e ring every endomorphism additive group multiplication element known notion theory abelian groups want change endomorphisms automorphisms define ring a ring every automorphism multiplication element many torsion free finite rank tffr a rings actually e rings while have example mixed a ring e ring still there tffr a rings e rings employ strong black box construct large integral domains a rings e rings