Decompositions of Monomial Ideals in Real Semigroup Rings

Abstract

Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this article, we investigate decompositions in the set of monomial ideals in the semigroup ring 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.

Key Words:

• Irreducible decompositions
• Monomial ideals
• Semigroup rings

2010 Mathematics Subject Classification:

• 13C05
• 13F20

ACKNOWLEDGMENTS

We are grateful to the anonymous referee for her/his thoughtful comments.

This material was supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. Sean Sather-Wagstaff was supported in part by a grant from the NSA.

Notes

One of the most interesting aspects of this theory is found in its interactions with combinatorics, including applications to graphs and simplicial complexes; see, e.g., [3-5, 9, 10, 15 17-19, 21, 22, 25, 27]. Foundational material on the subject can be found in the following texts [1, 13, 14, 20, 23, 26, 28].

Note that the case S = ∅ is covered by the convention that the empty sum of ideals is the zero ideal; the case [[R]]∖[[I]] = ∅ is similarly covered since the empty intersection contains the empty product which is the unit ideal.

Communicated by I. Swanson.