DOI QR코드

DOI QR Code

ON INJECTIVITY AND P-INJECTIVITY, IV

  • Published : 2003.05.01

Abstract

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).

References

  1. Rend. Sem. Mat. Univ. Padova v.72 Generalized V-rings and von Neumann regular rings G.Baccella
  2. Comm. Algebra v.23 Properly semi-prime self-pp-modules K.Beidar;R.Wisbauer https://doi.org/10.1080/00927879508825252
  3. AMS Math. Surveys and Monographs v.65 Rings and things and a fine array of twentieth century associative algebra C.Faith
  4. Ring Theory : Nonsingular rings and modules K.R.Goodearl
  5. Von Neumann regular rings K.R.Goodearl
  6. Publ. Math. v.38 On non-singular p-injective rings Y.Hirano https://doi.org/10.5565/PUBLMAT_38294_14
  7. London Math. Soc. Monographs v.17 no.C.U.P. Modules and rings F.Kasch
  8. Graduate Texts in Math. v.189 Lectures on modules and rings T.Y.Lam
  9. London Math. Soc. Lecture Note Series v.147 no.C.U.P. Continuous and discrete modules S.H.Mohammed;B.J.Mueller
  10. Journal of Algebra v.174 Principally injective rings W.K.Nicholson;M.F.Yousif https://doi.org/10.1006/jabr.1995.1117
  11. Glasgow Math. J. v.37 On p-injective rings G.Puninski;R.Wisbauer;M.F.Yousif https://doi.org/10.1017/S0017089500031657
  12. Foundations of module and ring theory R.Wisbauer
  13. Riv. Mat. Univ. Parma v.1 no.6 A note on YJ-injectivity WeiMin Xue
  14. Math. J. Okayama Univ. v.28 On SI-modules M. F.Yousif
  15. Glasgow Math. J. v.37 On quasi-duo rings HuaPing Yu https://doi.org/10.1017/S0017089500030342
  16. Proc. Edinburgh Math. Soc. v.19 On von Neumann regular rings R.Yue Chi Ming https://doi.org/10.1017/S0013091500015418
  17. Math. Japonica v.19 On simple p-injective modules R.Yue Chi Ming
  18. Math. Scandinavica v.39 On von Neumann regular rings Ⅱ R.Yue Chi Ming https://doi.org/10.7146/math.scand.a-11654
  19. Math. J. Okayama Univ. v.20 On generalizations of V-rings and regular rings R.Yue Chi Ming
  20. Rend. Sem. Mat. Univ. Torino v.39 On von Neumann regular rings, Ⅵ R.Yue Chi Ming
  21. Riv. Mat. Univ. Parma v.8 no.4 On regular rings and Artinian rings R.Yue Chi Ming
  22. Glasnik Mat. v.18 no.38 On von Neumann regular rings and self-injective rings, II R.Yue Chi Ming
  23. Annali di Mat. v.138 On von Neumann regular rings and continuous rings, Ⅲ R.Yue Chi Ming https://doi.org/10.1007/BF01762546
  24. Riv. Mat. Univ. Parma v.11 no.4 On regular rings and Artinian rings, Ⅱ R.Yue Chi Ming
  25. Ann. Univ. Fenara v.31 On von Neumann regular rings, XIII R.Yue Chi Ming https://doi.org/10.1007/BF02831758
  26. J. Math. Kyoto Uni. v.27 On injectivity and p-injectivity R.Yue Chi Ming https://doi.org/10.1215/kjm/1250520658
  27. Acta Math. Vietnamica v.13 On von Neumann regular rings, XV R.Yue Chi Ming
  28. Riv. Mat. Univ. Parma v.4 no.6 On injectivity and p-injectivity, III R.Yue Chi Ming
  29. Canad. J. Math. v.23 Injective hulls of torsionfree modules J.Zelmanowitz https://doi.org/10.4153/CJM-1971-115-x
  30. Algebra Colloquium v.6 Generalizations of principal injectvity Jule Zhang;Jun Wu

Cited by

  1. A Note on GP-Injectivity vol.16, pp.04, 2009, https://doi.org/10.1142/S1005386709000583