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nil−INJECTIVE RINGS

Year 2007, Volume: 2 Issue: 2, 1 - 21, 01.12.2007

Jun-chao Wei Jian-hua Chen

Abstract

A ring R is called left nil-injective if every R-homomorphism from a principal left ideal which is generated by a nilpotent element to R is a right multiplication by an element of R. In this paper, we first introduce and characterize a left nil-injective ring, which is a proper generalization of left p-injective ring. Next, various properties of left nil-injective rings are developed, many of them extend known results.

Keywords

Left minimal elements left min-abel rings strongly left min-abel rings left MC2 rings simple singular modules left nil−injective modules

References

  • J.L. Chen and N.Q. Ding. On generalizations of injectivity. In International Symposium on Ring Theory (Kyongju, 1999), 85-94, Birkhauser, Boston, Boston, MA, 2001.
  • J.L. Chen and N.Q. Ding. On regularity on rings. Algebra Colloq., 8 (2001), 267-274. [3] Jin Yong Kim. Certain rings whose simple singular modules are GP −injective. Proc. Japan Acad., 81, Ser. A (2005), 125-128.
  • N.K. Kim, S.B. Nam and J.K. Kim. On simple singular GP −injective modules. Comm. Algebra, 27(5)(1999), 2087-2096.
  • T.Y. Lam and Alex S. Dugas. Quasi-duo rings and stable range descent. J. Pure Appl. Algebra, 195 (2005), 243-259.
  • S. Lang and J.L. Chen. Small injective rings. arXiv:math.RA/0505445, vl:21 May 2005.
  • R. Yue Chi Ming. On Quasi-Frobeniusean and Artinian rings. Publications De L,institut Math´ematique, 33(47) (1983), 239-245
  • R.Yue Chi Ming. On p−injectivity and generalizations. Riv. Mat. Univ. Parma, (5)5 (1996), 183-188.
  • R. Yue Chi Ming. On quasi-injectivity and Von Neumann regularity. Mountshefte fr Math., 95 (1983), 25-32.
  • R. Yue Chi Ming. On Y J −injectivity and VNR rings. Bull. Math. Soc. Sc. Math. Roumanie Tome., 46(94)(1-2) (2003), 87-97.
  • G.O. Michler and O.E. Villamayor. On rings whose simple modules are injec- tive. J. Algebra, 25(1973), 185-201.
  • W.K. Nicholson and E.S. Campos. Rings with the dual of the isomorphism theorem. J. Algebra, 271 (2004), 391-406.
  • W.K. Nicholson and J.F. Watters. Rings with projective socle. Proc. Amer. Math. Soc. 102 (1988), 443-450.
  • W.K. Nicholson and M.F. Yousif. Minijective ring. J. Algebra, 187 (1997), 548-578. [15] W.K. Nicholson and M.F. Yousif. Weakly continuous and C2-rings. Comm. Algebra, 29(6) (2001), 2429-2466.
  • W.K. Nicholson and M.F. Yousif. Principally injective rings. J. Algebra, 174, 77-93 (1995), 77-93.
  • W.K. Nicholson and M.F. Yousif. On finitely embedded rings. Comm. Algebra, 28(11) (2000), 5311-5315.
  • W.K. Nicholson and M.F. Yousif. Continuous rings and chain conditions. J. Pure Appl. Algebra, 97 (1994), 325-332.
  • J.C. Wei. The rings characterized by minimal left ideal. Acta. Math. Sinica, 21(3) (2005), 473-482.
  • W. Xue. A note on Y J −injectivity. Riv. Mat. Univ. Parma, (6)1 (1998), 31-37. Jun-chao Wei*and Jian-hua Chen**
  • School of Mathematics Science, Yangzhou University,
  • Yangzhou,225002, Jiangsu, P. R. China
  • E-mail:*jcweiyz@yahoo.com.cn,**cjh_m@yahoo.com.cn

Year 2007, Volume: 2 Issue: 2, 1 - 21, 01.12.2007

Jun-chao Wei Jian-hua Chen

Abstract

References

  • J.L. Chen and N.Q. Ding. On generalizations of injectivity. In International Symposium on Ring Theory (Kyongju, 1999), 85-94, Birkhauser, Boston, Boston, MA, 2001.
  • J.L. Chen and N.Q. Ding. On regularity on rings. Algebra Colloq., 8 (2001), 267-274. [3] Jin Yong Kim. Certain rings whose simple singular modules are GP −injective. Proc. Japan Acad., 81, Ser. A (2005), 125-128.
  • N.K. Kim, S.B. Nam and J.K. Kim. On simple singular GP −injective modules. Comm. Algebra, 27(5)(1999), 2087-2096.
  • T.Y. Lam and Alex S. Dugas. Quasi-duo rings and stable range descent. J. Pure Appl. Algebra, 195 (2005), 243-259.
  • S. Lang and J.L. Chen. Small injective rings. arXiv:math.RA/0505445, vl:21 May 2005.
  • R. Yue Chi Ming. On Quasi-Frobeniusean and Artinian rings. Publications De L,institut Math´ematique, 33(47) (1983), 239-245
  • R.Yue Chi Ming. On p−injectivity and generalizations. Riv. Mat. Univ. Parma, (5)5 (1996), 183-188.
  • R. Yue Chi Ming. On quasi-injectivity and Von Neumann regularity. Mountshefte fr Math., 95 (1983), 25-32.
  • R. Yue Chi Ming. On Y J −injectivity and VNR rings. Bull. Math. Soc. Sc. Math. Roumanie Tome., 46(94)(1-2) (2003), 87-97.
  • G.O. Michler and O.E. Villamayor. On rings whose simple modules are injec- tive. J. Algebra, 25(1973), 185-201.
  • W.K. Nicholson and E.S. Campos. Rings with the dual of the isomorphism theorem. J. Algebra, 271 (2004), 391-406.
  • W.K. Nicholson and J.F. Watters. Rings with projective socle. Proc. Amer. Math. Soc. 102 (1988), 443-450.
  • W.K. Nicholson and M.F. Yousif. Minijective ring. J. Algebra, 187 (1997), 548-578. [15] W.K. Nicholson and M.F. Yousif. Weakly continuous and C2-rings. Comm. Algebra, 29(6) (2001), 2429-2466.
  • W.K. Nicholson and M.F. Yousif. Principally injective rings. J. Algebra, 174, 77-93 (1995), 77-93.
  • W.K. Nicholson and M.F. Yousif. On finitely embedded rings. Comm. Algebra, 28(11) (2000), 5311-5315.
  • W.K. Nicholson and M.F. Yousif. Continuous rings and chain conditions. J. Pure Appl. Algebra, 97 (1994), 325-332.
  • J.C. Wei. The rings characterized by minimal left ideal. Acta. Math. Sinica, 21(3) (2005), 473-482.
  • W. Xue. A note on Y J −injectivity. Riv. Mat. Univ. Parma, (6)1 (1998), 31-37. Jun-chao Wei*and Jian-hua Chen**
  • School of Mathematics Science, Yangzhou University,
  • Yangzhou,225002, Jiangsu, P. R. China
  • E-mail:*jcweiyz@yahoo.com.cn,**cjh_m@yahoo.com.cn
There are 21 citations in total.

Details

Other ID JA25CG73CG
Journal Section Articles
Authors

Jun-chao Wei This is me

Jian-hua Chen This is me

Publication Date December 1, 2007
Published in Issue Year 2007 Volume: 2 Issue: 2

Cite

APA Wei, J.-c., & Chen, J.-h. (2007). nil−INJECTIVE RINGS. International Electronic Journal of Algebra, 2(2), 1-21.
AMA Wei Jc, Chen Jh. nil−INJECTIVE RINGS. IEJA. December 2007;2(2):1-21.
Chicago Wei, Jun-chao, and Jian-hua Chen. “nil−INJECTIVE RINGS”. International Electronic Journal of Algebra 2, no. 2 (December 2007): 1-21.
EndNote Wei J-c, Chen J-h (December 1, 2007) nil−INJECTIVE RINGS. International Electronic Journal of Algebra 2 2 1–21.
IEEE J.-c. Wei and J.-h. Chen, “nil−INJECTIVE RINGS”, IEJA, vol. 2, no. 2, pp. 1–21, 2007.
ISNAD Wei, Jun-chao - Chen, Jian-hua. “nil−INJECTIVE RINGS”. International Electronic Journal of Algebra 2/2 (December 2007), 1-21.
JAMA Wei J-c, Chen J-h. nil−INJECTIVE RINGS. IEJA. 2007;2:1–21.
MLA Wei, Jun-chao and Jian-hua Chen. “nil−INJECTIVE RINGS”. International Electronic Journal of Algebra, vol. 2, no. 2, 2007, pp. 1-21.
Vancouver Wei J-c, Chen J-h. nil−INJECTIVE RINGS. IEJA. 2007;2(2):1-21.