Proceedings of the Japan Academy, Series A, Mathematical Sciences

Certain rings whose simple singular modules are GP-injective

Jin Yong Kim

Full-text: Open access

Abstract

We prove that if $R$ is an idempotent reflexive left Goldie ring whose simple singular left $R$-modules are GP-injective, then $R$ is a finite product of simple left Goldie rings. As a byproduct of this result we are able to show that if $R$ is semiprime, left Goldie and left weakly $\pi$-regular, then $R$ is a finite product of simple left Goldie rings.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 7 (2005), 125-128.

Dates
First available in Project Euclid: 3 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1128346015

Digital Object Identifier
doi:10.3792/pjaa.81.125

Mathematical Reviews number (MathSciNet)
MR2172601

Zentralblatt MATH identifier
1089.16004

Subjects
Primary: 16D50: Injective modules, self-injective rings [See also 16L60] 16E50: von Neumann regular rings and generalizations

Keywords
Generalized principally injective module idempotent reflexive ring simple singular module von Neumann regular ring Goldie ring

Citation

Kim, Jin Yong. Certain rings whose simple singular modules are GP -injective. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 7, 125--128. doi:10.3792/pjaa.81.125. https://projecteuclid.org/euclid.pja/1128346015


Export citation

References

  • G.F. Birkenmeier, J.Y. Kim and J.K. Park, A characterization of minimal prime ideals, Glasgow Math. J. 40 (1998), no. 2, 223–236.
  • A.W. Chatters and C.R. Hajarnavis, Rings with chain conditions, Pitman, Boston, Mass., 1980.
  • J. Chen and N. Ding, On generalizations of injectivity, in International Symposium on Ring Theory (Kyongju, 1999), 85–94, Birkhäuser, Boston, Boston, MA, 2001.
  • N.Q. Ding and J.L. Chen, Rings whose simple singular modules are YJ-injective, Math. Japon. 40 (1994), no. 1, 191–195.
  • C.Y. Hong, N.K. Kim, T.K Kwak and Y. Lee, On weak $\pi$-regularity of rings whose prime ideals are maximal, J. Pure Appl. Algebra 146 (2000), no. 1, 35–44.
  • N.K. Kim, S.B. Nam and J.Y. Kim, On simple singular GP-injective modules, Comm. Algebra 27 (1999), no. 5, 2087–2096.
  • G. Mason, Reflexive ideals, Comm. Algebra 9 (1981), no. 17, 1709–1724.
  • G.O. Michler and O.E. Villamayor, On rings whose simple modules are injective, J. Algebra 25 (1973), 185–201.
  • S.B. Nam, N.K. Kim and J.Y. Kim, On simple GP-injective modules, Comm. Algebra 23 (1995), no. 14, 5437–5444.
  • W.K. Nicholson and M.F. Yousif, Weakly continuous and C2-rings, Comm. Algebra 29 (2001), no. 6, 2429–2446.
  • W. Xue, A note on YJ-injectivity, Riv. Mat. Univ. Parma (6) 1 (1998), 31–37.
  • R. Yue Chi Ming, On von Neumann regular rings. III, Monatsh. Math. 86 (1978/79), no. 3, 251–257.
  • R. Yue Chi Ming, On regular rings and Artinian rings. II, Riv. Mat. Univ. Parma (4) 11 (1985), 101–109.
  • R. Yue Chi Ming, On $p$-injectivity and generalizations, Riv. Mat. Univ. Parma (5) 5 (1996), 183–188.
  • R. Yue Chi Ming, A note on YJ-injectivity, Demonstratio Math. 30 (1997), no. 3, 551–556.