Extensions of locally matricial and locally semisimple algebras
K. R. Goodearl
TL;DR
This paper addresses when local finiteness and semisimplicity are preserved under ring extensions for two classes of $K$-algebras: locally matricial algebras and locally finite dimensional semisimple algebras. It develops reduction techniques via regularity, corner analysis, and finite-dimensional subalgebra decompositions to prove that extensions of locally matricial algebras by locally matricial algebras remain locally matricial over any field, and that extensions of locally finite dimensional semisimple algebras by locally finite dimensional semisimple algebras are locally finite dimensional semisimple if and only if the base field is perfect (with counterexamples when not). Additionally, it shows that such extensions are always locally unit-regular, and it examines mixed and lifting conditions that govern these properties. The results delineate when local structural properties are stable under extensions and connect extension-closure to field perfection and regularity theory, with implications for ultramatricial and Chekanu-type phenomena in the landscape of regular rings. $K$-dependent phenomena and corner-lifting arguments are central to the methodology and conclusions.
Abstract
Two extension problems are solved. First, the class of locally matricial algebras over an arbitrary field is closed under extensions. Second, the class of locally finite dimensional semisimple algebras over a fixed field is closed under extensions if and only if the base field is perfect. Regardless of the base field, extensions of the latter type are always locally unit-regular.