Matrix wreath products of algebras and embedding theorems

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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 §6, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension ≥ 8 \geq 8 over a countable field which answers a question from [New trends in noncommutative algebra, Amer. Math. Soc., Providence, RI, 2012, pp. 41–52].