DOI QR코드

DOI QR Code

GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

  • Published : 2008.05.31

Abstract

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.

References

  1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852 https://doi.org/10.1080/00927879908826596
  2. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29 (2001), no. 2, 639-660 https://doi.org/10.1081/AGB-100001530
  3. W. E. Clark, Twisted matrix units semigroup algebras, Duck Math. J. 34 (1967), 417-423
  4. P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648 https://doi.org/10.1112/S0024609399006116
  5. J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32 (1990), 73-76
  6. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226 https://doi.org/10.1016/S0022-4049(99)00020-1
  7. C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122 https://doi.org/10.1081/AGB-120016752
  8. C. Y. Hong, N. K. Kim, and T. K. Kwak, Extensions of generalized reduced rings, Algebra Colloq. 12 (2005), no. 2, 229-240 https://doi.org/10.1142/S1005386705000222
  9. C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412 https://doi.org/10.1142/S1005386705000374
  10. I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968
  11. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223 https://doi.org/10.1016/S0022-4049(03)00109-9
  12. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300
  13. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368 https://doi.org/10.4153/CMB-1971-065-1
  14. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17
  15. G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60 https://doi.org/10.2307/1996398

Cited by

  1. ON QUASI-RIGID IDEALS AND RINGS vol.47, pp.2, 2010, https://doi.org/10.4134/BKMS.2010.47.2.385
  2. Generalized Semicommutative and Skew Armendariz Ideals vol.66, pp.9, 2015, https://doi.org/10.1007/s11253-015-1015-2
  3. Zero commutativity of nilpotent elements skewed by ring endomorphisms vol.45, pp.11, 2017, https://doi.org/10.1080/00927872.2017.1287267
  4. Zero Divisors in Skew Power Series Rings vol.43, pp.10, 2015, https://doi.org/10.1080/00927872.2014.946607
  5. STRUCTURE OF ZERO-DIVISORS IN SKEW POWER SERIES RINGS vol.52, pp.4, 2015, https://doi.org/10.4134/JKMS.2015.52.4.663
  6. QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS vol.26, pp.4, 2011, https://doi.org/10.4134/CKMS.2011.26.4.557
  7. INSERTION-OF-FACTORS-PROPERTY ON SKEW POLYNOMIAL RINGS vol.52, pp.6, 2015, https://doi.org/10.4134/JKMS.2015.52.6.1161