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.
Farid Kourki (Mon,) studied this question.