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# A Note on Algebras that are Sums of Two Subalgebras

Published:2016-02-11
Printed: Jun 2016
• Marek Kȩpczyk,
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15--351 Białystok, Poland
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## Abstract

We study an associative algebra $A$ over an arbitrary field, that is a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we generalize this result for semiprime algebras $A$. Consider the class of all semisimple finite dimensional algebras $A=B+C$ for some subalgebras $B$ and $C$ which satisfy given polynomial identities $f=0$ and $g=0$, respectively. We prove that all algebras in this class satisfy a common polynomial identity.
 Keywords: rings with polynomial identities, prime rings
 MSC Classifications: 16N40 - Nil and nilpotent radicals, sets, ideals, rings 16R10 - $T$-ideals, identities, varieties of rings and algebras 16S36 - Ordinary and skew polynomial rings and semigroup rings [See also 20M25] 16W60 - Valuations, completions, formal power series and related constructions [See also 13Jxx] 16R20 - Semiprime p.i. rings, rings embeddable in matrices over commutative rings

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