Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WHEN IS AN ENDOMORPHISM RING P-COHERENT?

  • Mao, Lixin (INSTITUTE OF MATHEMATICS NANJING INSTITUTE OF TECHNOLOGY)
  • Published : 2009.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

A ring is called left P-coherent if every principal left ideal is finitely presented. Let M be a right R-module with the endomorphism ring S. We mainly study the P-coherence of S. It is shown that S is a left P-coherent ring if and only if the left annihilator $ann_S$(X) is a finitely generated left ideal of S for any M-cyclic submodule X of M if and only if every cyclically M-presented right R-module has an M-torsionfree preenvelope. As applications, we investigate when the endomorphism ring S is left PP or von Neumann regular.

Keywords

References

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13. Springer-Verlag, New York-Heidelberg, 1974.
  2. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Mathematics, 30. Walter de Gruyter & Co., Berlin, 2000.
  3. E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, de Gruyter Expositions in Mathematics, 30. Walter de Gruyter & Co., Berlin, 2000.
  4. A. Hattori, A foundation of torsion theory for modules over general rings, Nagoya Math. J. 17 (1960), 147-158.
  5. S. Jondrup, p.p. rings and finitely generated flat ideals, Proc, Amer, Math, Soc. 28 (1971), 431-435.
  6. L. X. Mao, Relatively endocoherent modules, Algebra Colloq. 14 (2007), no. 2, 327-340. https://doi.org/10.1142/S1005386707000326
  7. L. X. Mao and N. Q. Ding, On divisible and torsionfree modules, Comm. Algebra 36 (2008), no. 2, 708-731. https://doi.org/10.1080/00927870701724201
  8. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  9. A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra 29 (2001), no. 5, 2039-2050. https://doi.org/10.1081/AGB-100002166
  10. B. Stenstrrom, Rings of Quotients, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
  11. R. B. Warfield, Jr., Purity and algebraic compactness for modules, Paciflc J. Math. 28 (1969), 699-719. https://doi.org/10.2140/pjm.1969.28.699
  12. W. Zimmermann, On locally pure-injective modules, J. Pure Appl. Algebra 166 (2002), no. 3, 337-357. https://doi.org/10.1016/S0022-4049(01)00011-1