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A matrix representation for associative algebras. I


Author: Jacques Lewin
Journal: Trans. Amer. Math. Soc. 188 (1974), 293-308
MSC: Primary 16A64; Secondary 16A42
DOI: https://doi.org/10.1090/S0002-9947-1974-0338081-5
MathSciNet review: 0338081
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Abstract: Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and $ \{ \delta (x),(x \in X)\} $ a basis for a free $ (F/U,F/V)$-bimodule T. Then the map $ x \to (\begin{array}{*{20}{c}} {x + V} & 0 \\ {\delta (x)} & {x + U} \\ \end{array} )$ induces an injective homomorphism $ F/UV \to (\begin{array}{*{20}{c}} {F/V} & 0 \\ T & {F/U} \\ \end{array} )$. If $ F/U$ and $ F/V$ are embeddable in matrices over a commutative algebra, so is $ F/UV$. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0338081-5
Keywords: Free algebra, universal derivation, embedding in matrices, matrix identities, Noetherian PI algebras, Abelian-by-nilpotent groups
Article copyright: © Copyright 1974 American Mathematical Society

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