Regular Ring Properties Degraded Through Inverse Limits

TL;DR

This paper explores how inverse limits can significantly weaken properties of regular rings, such as regularity and stable rank, suggesting they might be used to address longstanding algebraic problems.

Contribution

It demonstrates constructions where inverse limits degrade key properties of regular rings, providing new insights into their structural behavior.

Findings

Inverse limits can destroy regularity properties of rings.

Degradation of properties like unit-regularity and finite stable rank.

Potential implications for solving the Separativity Problem.

Abstract

We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to answer the long standing Separativity Problem (in the negative).