  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.
