On primary decompositions of unital locally matrix algebras

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.