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

Voir la notice de l'article provenant de la source Cambridge University Press

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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[1] [1] Amitsur, S. A., Nil Pi-rings. Proc. Amer. Math. Soc. 2(1951), 538–540. Google Scholar

[2] [2] Bahturin, Yu. and Giambruno, A., Identities of sums of commutative subalgebras. Rend. Circ. Mat. Palermo 43(1994), 250–258. Google Scholar | DOI

[3] [3] Beïdarand, K. I. Mikhalëv, A. V., Generalized polynomial identities and rings which are sums of two subrings. Algebra and Logic 34(1995), no. 1,1-5. Google Scholar | DOI

[4] [4] Felzenszwalb, B., Giambruno, A., and Leal, G., On rings which are sums of two Pl-subrings: a combinatorial approach. Pacific J. Math. 209(2003), no. 1,17-30. Google Scholar | DOI

[5] [5] Kegel, O. H., ZurNilpotenzgewisserassoziativerRinge. Math. Ann. 149(1962/1963), 258–260. Google Scholar | DOI

[6] [6] Kepczyk, M., Note on algebras which are sums of two PI subalgebras. J. Algebra Appl. 14(2015), 1550149, 10. Google Scholar | DOI

[7] [7] Kepczyk, M., Note on algebras that are sums of two subalgebras satisfying polynomial identities. British J. Math. Com. Sci. 4(2014), no. 23, 3245–3251. Google Scholar | DOI

[8] [8] Kepczyk, M., On algebras that are sums of two subalgebras satisfying certain polynomial identities. Publ. Math. Debrecen 72(2008), no. 3-4, 257–267. Google Scholar

[9] [9] Kepczyk, M. and Puczylowski, E. R., On radicals of rings which are sums of two subrings. Arch. Math. (Basel) 66(1996), 8–12. Google Scholar | DOI

[10] [10] Kepczyk, M. and Puczylowski, E. R., On the structure of rings which are sums of two subrings. Arch. Math. (Basel) 83(2004), no. 5, 429–436. Google Scholar | DOI

[11] [11] Kepczyk, M. and Puczylowski, E. R., Rings which are sums of two subrings. J. Pure Appl. Algebra 133(1998), no. 1-2, 151–162. Google Scholar | DOI

[12] [12] Kepczyk, M. and Puczylowski, E. R., Rings which are sums of two subrings satisfying polynomial identities.Comm. Algebra 29(2001), no. 5, 2059–2065. Google Scholar | DOI

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