The semitrivial ideal extension

Abstract

In this paper, we aim to generalize the ideal duplication defined for numerical semigroups to commutative rings with unity. We introduce the semitrivial ideal extension, a construction that, starting with an ideal of a commutative ring [Formula: see text] with unity and a submodule of a module [Formula: see text] over [Formula: see text], under specific assumptions, produces an ideal of the semitrivial extension [Formula: see text]. Using this tool we characterize the homogeneous prime ideals of a semitrivial extension and we completely describe the family of the maximal ideals. Furthermore, as it was done for the numerical duplication, using the semitrivial ideal extension, we characterize the modules [Formula: see text] such that [Formula: see text] is nearly Gorenstein.