Regular Ring Properties Degraded Through Inverse Limits

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).

Accepted for publication in Journal of Algebra. Minor edits + new section, developing a surjective inverse limit $Q=\underleftarrow{\lim }{Q}_i$ of regular rings such that $V(Q)$ is not isomorphic to $\underleftarrow{\lim }V(Q_i)$