CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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
Format:   LaTeX   MathJax   PDF  

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 rings with polynomial identities, prime rings
MSC Classifications: 16N40, 16R10, 16S36, 16W60, 16R20 show english descriptions Nil and nilpotent radicals, sets, ideals, rings
$T$-ideals, identities, varieties of rings and algebras
Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
Valuations, completions, formal power series and related constructions [See also 13Jxx]
Semiprime p.i. rings, rings embeddable in matrices over commutative rings
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
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/