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.