On primary decompositions of unital locally matrix algebras

TL;DR

This paper constructs a specific unital locally matrix algebra of uncountable dimension that defies primary decomposition and explores properties related to Steinitz numbers, providing counterexamples to existing questions.

Contribution

It introduces a unital locally matrix algebra with uncountable dimension that lacks a primary decomposition and exhibits unique Steinitz number properties.

Findings

Constructed a unital locally matrix algebra without primary decomposition.

Showed existence of algebras with arbitrary infinite Steinitz numbers.

Provided negative answers to prior open questions.

Abstract

We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from \cite{BezOl} and \cite{Kurochkin}. We also show that for an arbitrary infinite Steinitz number $s$ there exists a unital locally matrix algebra $A$ having the Steinitz number $s$ and not isomorphic to a tensor product of finite dimensional matrix algebras.