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ALMOST PRINCIPALLY SMALL INJECTIVE RINGS

  • Xiang, Yueming
  • Received : 2010.06.19
  • Published : 2011.11.01

Abstract

Let R be a ring and M a right R-module, S = $End_R$(M). The module M is called almost principally small injective (or APS-injective for short) if, for any a ${\in}$ J(R), there exists an S-submodule $X_a$ of M such that $l_Mr_R$(a) = Ma $Ma{\bigoplus}X_a$ as left S-modules. If $R_R$ is a APS-injective module, then we call R a right APS-injective ring. We develop, in this paper, APS-injective rings as a generalization of PS-injective rings and AP-injective rings. Many examples of APS-injective rings are listed. We also extend some results on PS-injective rings and AP-injective rings to APS-injective rings.

Keywords

APS-injective modules (rings);trivial extensions

References

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
  2. J. L. Chen and N. Q. Ding, On regularity of rings, Algebra Colloq. 8 (2001), no. 3, 267-274.
  3. J. L. Chen and Y. Q. Zhou, GP-injective rings need not be P-injective, Comm. Algebra 33 (2005), no. 7, 2395-2402. https://doi.org/10.1081/AGB-200058375
  4. J. L. Chen and Y. Q. Zhou, Extensions of injectivity and coherent rings, Comm. Algebra 34 (2006), no. 1, 275-288. https://doi.org/10.1080/00927870500346263
  5. K. Koike, Dual rings and cogenerator rings, Math. J. Okayama Univ. 37 (1995), 99-103.
  6. T. Y. Lam, A First Course in Noncommutative rings, Graduate Texts in Mathematic 131, Springer-Verlag, 2001.
  7. L. X. Mao, N. Q. Ding, and W. T. Tong, New characterizations and generalizations of PP-rings, Vietnam J. Math. 33 (2005), no. 1, 97-110.
  8. W. K. Nicholson and F. M. Yousif, Principally injective rings, J. Algebra 174 (1995), no. 1, 77-93. https://doi.org/10.1006/jabr.1995.1117
  9. W. K. Nicholson and F. M. Yousif, Mininjective rings, J. Algebra 187 (1997), no. 2, 548-578. https://doi.org/10.1006/jabr.1996.6796
  10. W. K. Nicholson and F. M. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  11. S. S. Page and Y. Q. Zhou, Generalizations of principally injective rings, J. Algebra 206 (1998), no. 2, 706-721. https://doi.org/10.1006/jabr.1998.7403
  12. L. Shen and J. Chen, New characterizations of quasi-Frobenius rings, Comm. Algebra 34 (2006), no. 6, 2157-2165. https://doi.org/10.1080/00927870600549667
  13. S. Wongwai, Almost mininjective rings, Thai J. Math. 4 (2006), no. 1, 245-249.
  14. Y. M. Xiang, Principally small injective rings, Kyungpook Math. J. 51 (2011), no. 2, 177-185.
  15. G. S. Xiao, Stable range and regularity of injective rings, (chinese) Ph. D thesis of Nanjing University, 2004.
  16. M. F. Yousif and Y. Q. Zhou, FP-injective, simple-injective, and quasi-Frobenius rings, Comm. Algebra 32 (2004), no. 6, 2273-2285. https://doi.org/10.1081/AGB-120037220
  17. Y. Q. Zhou, Rings in which certain right ideals are direct summands of annihilators, J. Aust. Math. Soc. 73 (2002), no. 3, 335-346. https://doi.org/10.1017/S1446788700009009