E-Rings and Related Structures

Abstract

An E-ring is a ring that is isomorphic to its ring of additive endomorphisms under the left regular representation. That is, a ring R is an E-ring provided R ≅ End(R+) under the map that sends r ∈ R to left multiplication by r. An R-module M is called an E-module over R if Hom _R (R, M) = Hom _Z (R, M). Despite their seemingly specialized definitions, E-rings, E-modules and related notions have played a major role in the theory of torsion-free abelian groups, and pop up with surprising frequency in other subject areas. Here are some examples:

• [19] A torsion-free abelian group G is cyclic and projective as a module over its endomorphism ring if and only if G = R ⊕ M, where R is an E-ring and M is an E-module over R.

• [27] A strongly indecomposable torsion-free abelian group G of finite rank is finitely generated over its endomorphism ring if and only if G is quasi-isomorphic to the additive group of an E-ring.

• [15] A strongly indecomposable torsion-free group G of finite rank is -uniserial as a module over its endomorphism ring only if G is a local, strongly homogeneous E-ring.

• [2] A universal algebra A is κ-free if there is a subset X of A of car dinality κ such that every function from X to A extends uniquely to an endomorphism of A. A ring that is l-free as an abelian group is precisely an E-ring (with X = {1}).