# Unital locally matrix algebras and Steinitz numbers

## TL;DR

This paper investigates unital locally matrix algebras over a field, introducing Steinitz numbers to classify them and exploring the relationship between these numbers and the algebraic structure.

## Contribution

It introduces Steinitz numbers as invariants for unital locally matrix algebras and analyzes their relationship with the algebra's structure.

## Key findings

- Steinitz numbers classify unital locally matrix algebras
- A relationship between Steinitz numbers and algebra structure is established
- Provides a framework for understanding the structure of locally matrix algebras

## Abstract

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 $A$.