Unital locally matrix algebras and Steinitz numbers

An $F$-algebra $A$ with unit $1$ is said to be a locally matrix algebra if an arbitrary finite collection of elements $a_1,$ $\ldots,$ $a_s$ from $A$ lies in a subalgebra $B$ with $1$ of the algebra $A$, that is isomorphic to a matrix algebra $M_n(F),$ $n\geq 1.$ To an arbitrary unital locally matrix algebra $A$ we assign a Steinitz number $\mathbf{n}(A)$ and study a relationship between $\mathbf{n}(A)$ and $A$.