Representable varieties of algebras
The author continues his study of representing algebras by matrix algebras that appeared in his articles [Algebra Logika 16, No.1, 3-23 (1977; Zbl 0381.16010) and Mat. Sb., Nov. Ser. 108(150), No.2, 168-186 (1979; Zbl 0401.16008)]. All algebras are associative algebras over a fixed field F. Theorem 1: Let R be an algebra satisfying the identities \([x_ 1,...,x_ n]\cdot y_ 1\cdot...\cdot y_ n[z_ 1,...,z_ n]=0\) and \([x_ 1,y_ 1]\cdot...\cdot [x_ n,y_ n]=0\). Then R is embeddable into the algebra of lower triangular \(m\times m\) matrices over a commutative ring K, where m depends only on n. Theorem 2: Let V be a variety of algebras over a field of characteristic 0. Then every finitely generated V-algebra is embeddable into a matrix algebra over a commutative ring iff V satisfies the identities in Theorem 1 for some n. The case of finite characteristic remains open.
representing algebras by matrix algebras
scientific article; zbMATH DE number 4137982