Integral degree of a ring and reduction numbers

TL;DR

This paper introduces the integral degree, a new invariant of rings, and explores its relationship with reduction numbers, providing bounds for algebraic invariants like Castelnuovo-Mumford regularity and Artin-Rees numbers.

Contribution

It defines the integral degree and establishes its role in bounding reduction numbers and related algebraic invariants.

Findings

Integral degree is finite for rings with finite integral closure.

Bounds for Castelnuovo-Mumford regularity of Rees algebra are derived.

Bounds for Artin-Rees numbers are obtained.

Abstract

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers.