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An F -algebra A with unit 1 is said to be a locally matrix algebra if an arbitrary finite collection of elements a 1 , ... , 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 ≥ 1. To an arbitrary unital locally matrix algebra A , we assign a Steinitz number s t (A) and study a relationship between s t (A) and A. [ABSTRACT FROM AUTHOR]

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 [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443-454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. 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. [ABSTRACT FROM AUTHOR]