A matrix representation for associative algebras. I

Lewin, Jacques

doi:10.1090/S0002-9947-1974-0338081-5

A matrix representation for associative algebras. I
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Trans. Amer. Math. Soc. 188 (1974), 293-308 Request permission

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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