Graded-irreducible modules are irreducible

TL;DR

This paper proves that graded-irreducible modules in Noetherian modules are irreducible and explores the properties of graded-irreducible decompositions, extending classical concepts to the graded setting.

Contribution

It establishes that graded-irreducibility implies irreducibility and extends the index of reducibility to graded modules, analyzing decomposition uniqueness and relationships with non-graded ideals.

Findings

Graded-irreducible modules are irreducible.

The index of reducibility extends naturally to graded modules.

Connections between graded and non-graded ideal decompositions.

Abstract

We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural extension of the index of reducibility to the graded setting coincides with the ordinary index of reducibility. We also investigate the question of uniqueness of the components in a graded-irreducible decomposition, as well as the relation between the index of reducibility of a non-graded ideal and that of its largest graded subideal.