On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-21-2023-108
Published: Oct 4, 2023
Cite as:
Eltiyeb Ali, On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings, Int. J. Anal. Appl., 21 (2023), 108.

Main Article Content

Eltiyeb Ali

Abstract

In this paper, we consider a ring R and a monoid M equipped with a twisting map f: M×M -> U(R) and an action map ω: M -> Aut(R). The main objective of our study is to investigate the conditions under which the crossed product structure R⋊M is p.q.-Baer and quasi-Baer rings, and how this property relates to the p.q.-Baer property of R and the existence of a generalized join in I(R) for M-indexed subsets, where I(R) denotes the set of ideals of R. Additionally, we prove a connection between R being a left p.q.-Baer ring and the CM-quasi-Armendariz property. Moreover, we prove that for any element φ2=φ, there exist an idempotent element e2=e such that φ = ce. We then prove that R is quasi-Baer if and only if the crossed product structure R⋊M is quasi-Baer. Finally, we present novel results regarding various constructions for crossed products.

Article Details

References

  1. I. Kaplansky, Rings of Operators, Benjamin, New York, (1968).
  2. W.E. Clark, Twisted Matrix Units Semigroup Algebras, Duke Math. J. 34 (1967), 417–423. https://doi.org/10.1215/s0012-7094-67-03446-1.
  3. G.F. Birkenmeier, J.Y. Kim, J.K. Park, Principally Quasi-Baer Rings, Commun. Algebra. 29 (2001), 639–660. https://doi.org/10.1081/agb-100001530.
  4. G.F. Birkenmeier, J.Y. Kim, J.K. Park, A Sheaf Representation of Quasi-Baer Rings, J. Pure Appl. Algebra. 146 (2000), 209–223. https://doi.org/10.1016/s0022-4049(99)00164-4.
  5. G.F. Birkenmeier, J.Y. Kim, J.K. Park, On quasi-Baer rings, in: D.V. Huynh, S.K. Jain, S.R. Lopez-Permouth (Eds.), Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 2000: pp. 67–92. https://doi.org/10.1090/conm/259/04088.
  6. G.F. Birkenmeier, J.Y. Kim, J.K. Park, On Polynomial Extensions of Principally Quasi-Baer Rings, Kyungpook Math. J. 40 (2000), 247–253.
  7. E. Ali, The Reflexive Condition on Skew Monoid Rings, Eur. J. Pure Appl. Math. 16 (2023), 1878–1893.
  8. Z. Liu, a Note on Principally Quasi-Baer Rings, Commun. Algebra. 30 (2002), 3885–3890. https://doi.org/10.1081/agb-120005825.
  9. L. Zhao, Y. Zhou, Generalised Armendariz Properties of Crossed Product Type, Glasgow Math. J. 58 (2015), 313–323. https://doi.org/10.1017/s001708951500021x.
  10. E. Ali, Generalized Reflexive Structures Properties of Crossed Products Type, Eur. J. Pure Appl. Math. In Press.
  11. A.V. Kelarev, Ring Constructions and Applications, World Scientific, 2001. https://doi.org/10.1142/4807.
  12. D.S. Passman, The Algebraic Structure of Group Rings, John Wiley, Sons Ltd., New York, 1977.
  13. Y. Hirano, on Ordered Monoid Rings Over a Quasi-Baer Ring, Commun. Algebra. 29 (2001), 2089–2095. https://doi.org/10.1081/agb-100002171.
  14. A.R. Nasr-Isfahani, A. Moussavi, on Weakly Rigid Rings, Glasgow Math. J. 51 (2009), 425–440. https://doi.org/10.1017/s0017089509005084.
  15. E. Hashemi, A. Moussavi, Polynomial Extensions of Quasi-Baer Rings, Acta Math. Hung. 107 (2005), 207–224. https://doi.org/10.1007/s10474-005-0191-1.
  16. G.F. Birkenmeier, J.K. Park, Triangular Matrix Representations of Ring Extensions, J. Algebra. 265 (2003), 457– 477. https://doi.org/10.1016/s0021-8693(03)00155-8.
  17. Z.K. Liu, Quasi-Baer Rings of Generalized Power Series, Chinese Ann. Math. 23 (2002), 579–584.
  18. Z. Liu, J. Ahsan, PP-Rings of Generalized Power Series, Acta Math. Sinica. 16 (2000), 573–578. https://doi.org/10.1007/s1011400000884.
  19. Z.K. Liu, Principal Quasi-Baerness of Rings of Generalized Power Series, Northeast Math. J. 23 (2007), 283–292.
  20. R. Mazurek, Left Principally Quasi-Baer and Left APP-Rings of Skew Generalized Power Series, J. Algebra Appl. 14 (2014), 1550038. https://doi.org/10.1142/s0219498815500383.
  21. E. Ali, A. Elshokry, Some Properties of Quasi-Armendariz Rings and Their Generalization, Asia Pac. J. Math. 5 (2018), 14-26.
  22. L. Zhongkui, Z. Wenhui, Quasi-Armendariz Rings Relative to a Monoid, Commun. Algebra. 36 (2008), 928–947. https://doi.org/10.1080/00927870701776698.
  23. E. Ali, A. Elshokry, A Note on (S, ω)-Quasi-Armendariz Rings, Palestine J. Math. In Press.
  24. L. Zhongkui, Z. Renyu, a Generalization of pp-Rings and p.q.-Baer Rings, Glasgow Math. J. 48 (2006), 217–229. https://doi.org/10.1017/s0017089506003016.