Regular modules and homogeneous trivial extensions

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.