Representable varieties of algebras
type: Item
instance of: scholarly article
title: Representable varieties of algebras
zbMATH Open document ID: 0695.16013
DOI: 10.1007/BF01979372
author: A. Z. Anan'in
published in: Algebra and Logic
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 4137982
zbMATH Keywords: representing algebras by matrix algebras
zbMATH Keywords: identities
zbMATH Keywords: variety of algebras
zbMATH Keywords: embeddable into a matrix algebra
reviewed by: Awad A. Iskander
MaRDI profile type: Publication
cites work: Q5834703
cites work: Q4161815
cites work: Q3916700
description: scientific article; zbMATH DE number 4137982