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This is a Quantitative Study study.

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November 4, 2025

Regular modules and homogeneous trivial extensions

Key Points

Characterization of H-regular modules involves integral domains and specific ideal forms.
An integral domain condition is outlined for H-regular modules within finitely generated modules.
Analysis of module types focuses on trivial extensions and their homogeneous nature.
Implications for understanding various regularity concepts in ring theory are highlighted.

Abstract

Let Formula: see text be a commutative ring and Formula: see text an Formula: see text-module. We call Formula: see text H-regular if the trivial extension Formula: see text is homogeneous, that is, every ideal of Formula: see text is of the form Formula: see text where Formula: see text is an ideal of Formula: see text and Formula: see text is a submodule of Formula: see text. Various characterizations and properties of H-regular modules are given. In particular, we show that Formula: see text is H-regular if and only if for every maximal ideal Formula: see text of Formula: see text such that Formula: see text, Formula: see text is an integral domain and Formula: see text is a divisible Formula: see text-module. In the case Formula: see text is a nonzero finitely generated module, we find that Formula: see text is H-regular if and only if Formula: see text is a von Neumann regular ring and Formula: see text is a pure ideal of Formula: see text. Further, we investigate the relations between H-regular modules and other notions of regularity. Several examples are provided to illustrate and delimit the results obtained.

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