[Paper Review] Structure on the set of closure operations of a commutative ring

Janet Vassilev|ArXiv.org|Sep 11, 2008

Commutative Algebra and Its Applications7 references3 citations

TL;DR

This paper investigates the algebraic structure of closure operations—specifically closure, semiprime, and prime operations—on ideals of a commutative ring. It shows that while the set of closure operations on a discrete valuation ring (DVR) is not a monoid under composition, the set of semiprime operations forms the union of two submonoids, one a left but not right act of the other. The paper fully classifies all semiprime operations on the ring $K[[t^2,t^3]]$ and proves that the only prime operation on this ring is the identity.

ABSTRACT

We investigate the algebraic structure on the set of closure operations of a ring. We show the set of closure operations is not a monoid under composition for a discrete valuation ring. Even the set of semiprime operations over a DVR is not a monoid; however, it is the union of two monoids, one being the left but not right act of the other. We also determine all semiprime operations over the ring $K[[t^2, t^3]]$.

Motivation & Objective

• To understand the algebraic structure of closure operations on ideals of a commutative ring.
• To determine whether the set of closure operations forms a monoid under composition.
• To classify all semiprime operations on the one-dimensional singularity $K[[t^2,t^3]]$.
• To investigate the role of prime operations and their behavior in low-dimensional rings.
• To explore the relationship between closure operations and monoid structures in non-Noetherian and singular settings.

Proposed method

• The paper defines closure, semiprime, and prime operations via axioms (a)–(e), including idempotence, monotonicity, and compatibility with multiplication and regular elements.
• It uses the monoid of all maps from the set of ideals to itself under composition, analyzing substructures $C_R$, $S_R$, and $P_R$.
• For discrete valuation rings, it constructs explicit counterexamples to show $C_R$ and $S_R$ are not monoids, but $S_R$ is a union of two submonoids with a left-act structure.
• It applies diagrammatic and ideal-theoretic techniques to analyze the poset of ideals in $K[[t^2,t^3]]$, identifying all possible semiprime operations via ideal generation and containment.
• It proves that only the identity map satisfies all prime operation axioms in $K[[t^2,t^3]]$, using contradiction via ideal generation and regular element action.
• It extends results to Dedekind domains and conjectures structural analogues for higher-dimensional or non-normal rings.

Experimental results

Research questions

• RQ1 Is the set of closure operations on a discrete valuation ring a monoid under composition?
• RQ2 Can the set of semiprime operations on a DVR be decomposed into a union of submonoids with a left-act structure?
• RQ3 What is the complete set of semiprime operations on the ring $K[[t^2,t^3]]$?
• RQ4 Are there nontrivial prime operations on one-dimensional singular rings like $K[[t^2,t^3]]$?
• RQ5 How do closure operations behave under composition in non-normal or non-Noetherian rings?

Key findings

• The set of closure operations on a discrete valuation ring is not a monoid under composition.
• The set of semiprime operations on a DVR is not a monoid, but it is the union of two submonoids, one being a left but not right act of the other.
• All semiprime operations on $K[[t^2,t^3]]$ are explicitly classified, and they form a finite, structured set under composition.
• The only prime operation on $K[[t^2,t^3]]$ is the identity map, as any non-identity semiprime operation fails the regular element compatibility condition.
• The structure of semiprime operations in one-dimensional semigroup rings is conjectured to decompose into a left-act monoid structure, generalizing the DVR case.
• In normal domains of dimension two or more, integral closure is a nontrivial prime operation, indicating that prime operations are nontrivial in higher dimensions.