On primary decompositions of unital locally matrix algebras
Oksana Bezushchak, Bogdana Oliynyk

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 there exists a unital locally matrix algebra having the Steinitz number and not isomorphic to a tensor product of finite dimensional matrix algebras.
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