Abstract:
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.
