On the matrix wreath product of algebras

Abstract

We study the singular ideal, the Jacobson radical and the maximal quotient rings of the matrix wreath product of two algebras and certain subrings as defined in (Alamhadi-Jain-Alsulami and Zelmanov, Electron. Res. Announc. Math. Sci. (2017)) [1]. We give formulas for computing the singular ideal and the Jacobson radical of a matrix wreath product. We also give the structure of the maximal right quotient ring of the infinite matrix ring of a right nonsingular algebra and, as consequence, we give the maximal quotient ring of a matrix wreath product. As an application of the fact that matrix wreath product of nonsingular algebras is nonsingular we construct another algebra (similar to the one in [3]) showing that any nonsingular algebra of countable dimension can be embedded in a finitely generated nonsingular algebra.