Modules Which are Isomorphic to Their Factor Modules

Let R be commutative ring with identity and let M be an infinite unitary R-module.

Call M homomorphically congruent (HC for short) provided M/N M for every

submodule N of M for which M/N=M. In this article, we study HC modules over

commutative rings. After a fairly comprehensive review of the literature, several natural

examples are presented to motivate our study. We then prove some general results

on HC modules, including HC module-theoretic characterizations of discrete valuation

rings, almost Dedekind domains, and fields. We also provide a characterization of the

HC modules over a Dedekind domain, extending Scott’s classification over in [22].

Finally, we close with some open questions.