Geodesic
A Note on Algebras that are Sums of Two Subalgebras
Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 340-345

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DOI

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.
DOI : 10.4153/CMB-2015-082-6
Mots-clés : 16N40, 16R10, 16S36, 16W60, 16R20, rings with polynomial identities, prime rings 
Kępczyk, Marek. A Note on Algebras that are Sums of Two Subalgebras. Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 340-345. doi: 10.4153/CMB-2015-082-6
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     author = {K\k{e}pczyk, Marek},
     title = {A {Note} on {Algebras} that are {Sums} of {Two} {Subalgebras}},
     journal = {Canadian mathematical bulletin},
     pages = {340--345},
     year = {2016},
     volume = {59},
     number = {2},
     doi = {10.4153/CMB-2015-082-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2015-082-6/}
}
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