Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON CONDITIONS PROVIDED BY NILRADICALS

  • Kim, Hong-Kee (DEPARTMENT OF MATHEMATICS GYEONGSANG NATIONAL UNIVERSITY) ;
  • Kim, Nam-Kyun (DIVISION OF GENERAL EDUCATION HANBAT NATIONAL UNIVERSITY) ;
  • Jeong, Mun-Seob (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY) ;
  • Lee, Yang (DEPARTMENT OF MATHEMATICS EDUCATION BUSAN NATIONAL UNIVERSITY) ;
  • Ryu, Sung-Ju (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY) ;
  • Yeo, Dong-Eun (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY)
  • Published : 2009.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

A ring R is called IFP, due to Bell, if ab = 0 implies aRb = 0 for a, b $\in$ R. Huh et al. showed that the IFP condition is not preserved by polynomial ring extensions. In this note we concentrate on a generalized condition of the IFPness that can be lifted up to polynomial rings, introducing the concept of quasi-IFP rings. The structure of quasi-IFP rings will be studied, characterizing quasi-IFP rings via minimal strongly prime ideals. The connections between quasi-IFP rings and related concepts are also observed in various situations, constructing necessary examples in the process. The structure of minimal noncommutative (quasi-)IFP rings is also observed.

Keywords

References

  1. A. Badawi, On abelian $\pi$-regular rings, Comm. Algebra 25 (1997), no. 4, 1009.1021 https://doi.org/10.1080/00927879708825906
  2. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363.368 https://doi.org/10.1017/S0004972700042052
  3. G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, Completely prime ideals and associated radicals, Ring theory (Granville, OH, 1992), 102.129, World Sci. Publ., River Edge, NJ, 1993
  4. A. W. Chatters and C. R. Hajarnavis, Rings with Chain Conditions, Research Notes in Mathematics, 44. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980
  5. N. J. Divinsky, Rings and Radicals, Mathematical Expositions No. 14 University of Toronto Press, Toronto, Ont. 1965
  6. D. B. Erickson, Orders for finite noncommutative rings, Amer. Math. Monthly 73 (1966), 376.377 https://doi.org/10.2307/2315402
  7. K. R. Goodearl, von Neumann Regular Rings, Monographs and Studies in Mathematics, 4. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979
  8. K.-Y. Ham, Y. C. Jeon, J. Kang, N. K. Kim, W. Lee, Y. Lee, S. J. Ryu, and H.-H. Yang, IFP Rings and Near-IFP Rings, J. Korean Math. Soc. 45 (2008), no. 3, 727.740 https://doi.org/10.4134/JKMS.2008.45.3.727
  9. C. Y. Hong and T. K. Kwak, On minimal strongly prime ideals, Comm. Algebra 28 (2000), no. 10, 4867.4878 https://doi.org/10.1080/00927870008827127
  10. C. Huh, H. K. Kim, and Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37.52 https://doi.org/10.1016/S0022-4049(01)00149-9
  11. C. Huh, N. K. Kim, and Y. Lee, Examples of strongly $\pi$-regular rings, J. Pure Appl. Algebra 189 (2004), no. 1-3, 195.210 https://doi.org/10.1016/j.jpaa.2003.10.032
  12. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751.761 https://doi.org/10.1081/AGB-120013179
  13. S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186.199 https://doi.org/10.1016/j.jalgebra.2006.02.032
  14. S. U. Hwang, N. K. Kim, and Y. Lee, On rings whose right annihilators are bounded, Glasgow Math. J., To appear https://doi.org/10.1017/S0017089509005163
  15. Y. C. Jeon, H. K. Kim, and J. S. Yoon, On weak Armendariz rings, Bull. Korean Math. Soc. 46 (2009), no. 1, 135.146 https://doi.org/10.4134/BKMS.2009.46.1.135
  16. 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
  17. N. K. Kim, Y. Lee, and S. J. Ryu, An ascending chain condition on Wedderburn radicals, Comm. Algebra 34 (2006), no. 1, 37.50 https://doi.org/10.1080/00927870500345901
  18. 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
  19. C. Lanski, Nil subrings of Goldie rings are nilpotent, Canad. J. Math. 21 (1969), 904. 907 https://doi.org/10.4153/CJM-1969-098-x
  20. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113.2123 https://doi.org/10.1081/AGB-100002173
  21. L. Motais de Narbonne, Anneaux semi-commutatifs et $unis{\acute{e}}riels$; anneaux dont les $id{\acute{e}}aux$ principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71.73, Bib. Nat., Paris, 1982
  22. L. H. Rowen, Ring Theory, Academic Press, Inc., Boston, MA, 1991
  23. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43.60 https://doi.org/10.1090/S0002-9947-1973-0338058-9
  24. W. Xue, On strongly right bounded finite rings, Bull. Austral. Math. Soc. 44 (1991), no. 3, 353.355 https://doi.org/10.1017/S000497270002983X
  25. W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded nonduo rings, Comm. Algebra 20 (1992), no. 9, 2777.2788 https://doi.org/10.1080/00927879208824488

Cited by

  1. On Commutativity of Semiprime Right Goldie C<i><sub>k</sub></i>-Rings vol.02, pp.04, 2012, https://doi.org/10.4236/apm.2012.24031
  2. ON WEAK ZIP SKEW POLYNOMIAL RINGS vol.05, pp.03, 2012, https://doi.org/10.1142/S1793557112500398
  3. Mal’cev-Neumann series over rings satisfy the weak Beachy–Blair condition 2016, https://doi.org/10.1007/s12215-016-0262-x
  4. NC-Rings and Some Commutativity Conditions vol.09, pp.02, 2019, https://doi.org/10.4236/apm.2019.92008