Modules Which are Isomorphic to Their Factor Modules

Abstract

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 ℤ in [22]. Finally, we close with some open questions.

Key Words:

• Almost Dedekind domain
• Artinian module
• Dedekind domain
• Discrete valuation ring
• Noetherian module
• Strong limit cardinal
• Socle
• Uniserial module

2010 Mathematics Subject Classification:

• Primary: 13A99
• 13C05
• Secondary: 13E05
• 03E50

Notes

Communicated by I. Swanson.