Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T20:10:13.541Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1] Amitsur, S. A., Nil Pi-rings. Proc. Amer. Math. Soc. 2(1951), 538–540.Google Scholar
[2] Bahturin, Yu. and Giambruno, A., Identities of sums of commutative subalgebras. Rend. Circ. Mat. Palermo 43(1994), 250–258. http://dx.doi.org/10.1007/BF02844841 Google Scholar
[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. http://dx.doi.org/10.1007/BF00750549 Google Scholar
[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. http://dx.doi.org/10.2140/pjm.2003.209.17 Google Scholar
[5] Kegel, O. H., ZurNilpotenzgewisserassoziativerRinge. Math. Ann. 149(1962/1963), 258–260. http://dx.doi.org/10.1007/BF01470877 Google Scholar
[6] Kepczyk, M., Note on algebras which are sums of two PI subalgebras. J. Algebra Appl. 14(2015), 1550149, 10.http://dx.doi.org/10.1142/S0219498815501492 Google Scholar
[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.http://dx.doi.org/10.9734/BJMCS/2014/12652 Google Scholar
[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] Kepczyk, M. and Puczylowski, E. R., On radicals of rings which are sums of two subrings. Arch. Math. (Basel) 66(1996), 8–12. http://dx.doi.org/10.1007/BF01323977 Google Scholar
[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. http://dx.doi.org/10.1007/s00013-004-1206-9 Google Scholar
[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. http://dx.doi.org/10.1016/S0022-4049(97)00191-6 Google Scholar
[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. http://dx.doi.org/10.1081/ACB-100002168 Google Scholar