Decompositions of Monomial Ideals in Real Semigroup Rings

TL;DR

This paper extends the theory of monomial ideal decompositions from polynomial rings over fields to more general semigroup rings over arbitrary rings, classifying irreducible elements and those with finite decompositions.

Contribution

It introduces a classification of m-irreducible monomial ideals and characterizes ideals that can be finitely decomposed in the semigroup ring setting.

Findings

Classification of m-irreducible monomial ideals

Characterization of ideals with finite decompositions

Extension of decomposition theory to real semigroup rings

Abstract

Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an arbitrary commutative ring with identity. We classify the irreducible elements of this set, which we call m-irreducible, and we classify the elements that admit decompositions into finite intersections of m-irreducible ideals.