Abstract view
A Note on Algebras that are Sums of Two Subalgebras


Published:20160211
Printed: Jun 2016
Marek Kȩpczyk,
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15351 Białystok, Poland
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.
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
