Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On irreducible modules over $ q$-skew polynomial rings and smash products


Author: Piotr Grzeszczuk
Journal: Proc. Amer. Math. Soc. 142 (2014), 59-69
MSC (2010): Primary 16N20, 16S36, 16W25, 16S40
DOI: https://doi.org/10.1090/S0002-9939-2013-11794-6
Published electronically: September 12, 2013
MathSciNet review: 3119181
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be an irreducible left module over a $ q$-skew polynomial ring $ R[x;\sigma ,\delta ]$. We give sufficient conditions for the complete reducibility of $ M$ considered as a module over the coefficient ring $ R$. We apply it to irreducible modules over smash product $ R\char93 H$, where $ H$ is a Hopf algebra generated by skew primitive elements.


References [Enhancements On Off] (What's this?)

  • [1] Carl Faith, Algebra. II. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin, 1976. MR 0427349 (55 #383)
  • [2] Edward Formanek and Arun Vinayak Jategaonkar, Subrings of Noetherian rings, Proc. Amer. Math. Soc. 46 (1974), 181-186. MR 0414625 (54 #2725)
  • [3] K. R. Goodearl and E. S. Letzter, Prime ideals in skew and $ q$-skew polynomial rings, Mem. Amer. Math. Soc. 109 (1994), no. 521, vi+106. MR 1197519 (94j:16051)
  • [4] P. Grzeszczuk, On $ G$-systems and $ G$-graded rings, Proc. Amer. Math. Soc. 95 (1985), no. 3, 348-352. MR 806068 (87a:16003), https://doi.org/10.2307/2045800
  • [5] V. K. Kharchenko, An algebra of relatively primitive elements, Algebra i Logika 37 (1998), no. 2, 181-223, 249 (Russian, with Russian summary); English transl., Algebra and Logic 37 (1998), no. 2, 101-126. MR 1672889 (99m:16061), https://doi.org/10.1007/BF02671596
  • [6] V. Linchenko, S. Montgomery, and L. W. Small, Stable Jacobson radicals and semiprime smash products, Bull. London Math. Soc. 37 (2005), no. 6, 860-872. MR 2186719 (2006k:16084), https://doi.org/10.1112/S0024609305004662
  • [7] Maria E. Lorenz and Martin Lorenz, Observations on crossed products and invariants of Hopf algebras, Arch. Math. (Basel) 63 (1994), no. 2, 119-127. MR 1289292 (95f:16036), https://doi.org/10.1007/BF01189884
  • [8] Martin Lorenz and D. S. Passman, Observations on crossed products and fixed rings, Comm. Algebra 8 (1980), no. 8, 743-779. MR 566420 (81c:16013), https://doi.org/10.1080/00927878008822488

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16N20, 16S36, 16W25, 16S40

Retrieve articles in all journals with MSC (2010): 16N20, 16S36, 16W25, 16S40


Additional Information

Piotr Grzeszczuk
Affiliation: Faculty of Computer Science, Białystok University of Technology, Wiejska 45A, 15-351 Białystok, Poland
Email: piotrgr@pb.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2013-11794-6
Keywords: Irreducible module, skew polynomial ring, skew derivation
Received by editor(s): November 19, 2011
Received by editor(s) in revised form: February 26, 2012
Published electronically: September 12, 2013
Additional Notes: The author was supported in part by the grant MNiSW nr N N201 268435 and by the grant S/WI/1/2011 of Białystok University of Technology.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society