  We introduce a new construction of matrix wreath products of algebras that is
similar to wreath products of groups. We then use it to prove embedding
theorems for Jacobson radical, nil, and primitive algebras. In
\S\ref{Section6}, we construct finitely generated nil algebras of arbitrary
Gelfand-Kirillov dimension $\geq 8$ over a countable field which answers a
question from \cite{8}.
