Demotions of ideals in commutative rings with applications to normally torsion-freeness

TL;DR

The paper introduces demotion of ideals as a compatibility condition $I^rJ^s=I^{r+s}\cap J^s$ for all $r,s\ge0$ and develops a comprehensive study in polynomial rings. It provides local characterizations, identifies monomial classes with the property (notably monomial primes and principal monomial ideals), and analyzes behavior under a wide array of monomial operations (summation, localization, contraction, deletion, permutation, expansion, weighting). It forges a strong link between demotions and normally torsion-free monomial ideals, showing how demotions can generate new NTF examples and how NTF structure informs demotion construction. Finally, it contrasts reductions and demotions, giving several counterexamples to illustrate fundamental differences between these two concepts.

Abstract

Let J \subseteq I be ideals in a commutative Noetherian ring R, and r,s \geq 0. We say that J is a demotion of I if I^r J^s = I^{r+s} \cap J^s for all r,s \geq 0. In this paper, we mainly aim to explore this notion in polynomial rings. In particular, we investigate the relation between the demotion property and normal torsion-freeness. Furthermore, we compare the reductions of ideals and demotions of ideals.

Theorems & Definitions (59)

• Theorem 2.5
• Proposition 2.6
• Theorem 2.7
• Lemma 2.8
• Lemma 2.10
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 4.1