Primary decompositions of unital locally matrix algebras
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.
Locally matrix algebra, primary decomposition, Steinitz number, tensor product, Clifford algebra, article
Bezushchak O. Primary decompositions of unital locally matrix algebras [electronic resource] / Bezushchak O., Oliynyk B. // Bulletin of Mathematical Sciences. - 2020. - Vol. 10, Issue 1. - Article number 2050006.