An upper bound on the reduction number of an ideal

TL;DR

This paper establishes an upper bound on the reduction number of an ideal in a commutative ring, relating it to properties of containing ideals and their generators, with implications for ideal powers.

Contribution

It provides a new upper bound on the reduction number based on a family of ideals and relates it to the structure of containing ideals satisfying specific algebraic conditions.

Findings

Upper bound on reduction number established

Relation between ideal powers and generators shown

Corollary on ideal power equality derived

Abstract

Let A be a commutative ring and I an ideal of A with a reduction Q. In this paper we give an upper bound on the reduction number of I with respect to Q, when a suitable family of ideals in A is given. As a corollary it follows that if some ideal J containing I satisfies J^2 = QJ, then I^{v + 2} = QI^{v + 1}, where v denotes the number of generators of J / I as an A-module.