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A Note on Algebras that are Sums of Two Subalgebras

Published online by Cambridge University Press:  20 November 2018

Marek Kępczyk
Marek Kępczyk*
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Białystok, Poland e-mail: m.kepczyk@pb.edu.pl
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Abstract

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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 $C$ that satisfy given polynomial identities $f\,=\,0$ and $g\,=\,0$ , respectively. We prove that all algebras in this class satisfy a common polynomial identity.

Keywords

16N4016R1016S3616W6016R20rings with polynomial identitiesprime rings
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 340 - 345
Copyright
Copyright © Canadian Mathematical Society 2016

References

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