Projectively full ideals in Noetherian rings, a survey

TL;DR

This survey explores the concept of projectively full ideals in Noetherian rings, examining their properties, connections with Rees valuations, and the effects of integral extensions on their existence.

Contribution

It provides a comprehensive overview of projectively full ideals, highlighting their characteristics, related valuation theory, and conditions for their existence or failure in Noetherian rings.

Findings

Identifies conditions for the existence of projectively full ideals.

Explores the relationship between Rees valuations and projective equivalence.

Discusses how integral extensions can influence ideal properties.

Abstract

We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question of whether improvements can be made by passing to an integral extension ring.