Taiwanese Journal of Mathematics

MODULES CHARACTERIZED BY THEIR SIMPLE SUBMODULES

Lixin Mao

Full-text: Open access

Abstract

$M$ is said to be a min-coherent (resp. $PS$, $FS$) module if its every simple submodule is finitely presented (resp. projective, flat). In this article, we study the properties of min-coherent, $PS$ and $FS$ modules. Some known results are generalized.

Article information

Source
Taiwanese J. Math., Volume 15, Number 5 (2011), 2337-2349.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406438

Digital Object Identifier
doi:10.11650/twjm/1500406438

Mathematical Reviews number (MathSciNet)
MR2880408

Zentralblatt MATH identifier
1241.16002

Subjects
Primary: 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 16D40: Free, projective, and flat modules and ideals [See also 19A13] 16D50: Injective modules, self-injective rings [See also 16L60]

Keywords
min-coherent module $PS$ module $FS$ module $M$-min-flat module $M$-min-injective module preenvelope precover

Citation

Mao, Lixin. MODULES CHARACTERIZED BY THEIR SIMPLE SUBMODULES. Taiwanese J. Math. 15 (2011), no. 5, 2337--2349. doi:10.11650/twjm/1500406438. https://projecteuclid.org/euclid.twjm/1500406438


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References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
  • V. Camillo, Coherence for polynomial rings, J. Algebra, 132 (1990), 72-76.
  • E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
  • J. R. Garc\ipzz a Rozas and B. Torrecillas, Relative injective covers, Comm. Algebra, 22 (1994), 2925-2940.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math. 1371, Springer-Verlag, 1989.
  • H. Holm and P. Jørgensen, Covers, precovers, and purity, Illinois J. Math., 52 (2008), 691-703.
  • T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York-Heidelberg-Berlin, 1999.
  • Z. K. Liu, Rings with flat left socle, Comm. Algebra, 23 (1995), 1645-1656.
  • Z. K. Liu, $PS$ modules over rings of generalized power series, Northeast. Math. J., 18(3) (2002), 254-260.
  • L. X. Mao, Min-flat modules and min-coherent rings, Comm. Algebra, 35(2) (2007), 635-650.
  • L. X. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen, 72(3-4) (2008), 347-358.
  • W. K. Nicholson and J. F. Watters, Rings with projective socle, Proc. Amer. Math. Soc., 102 (1988), 443-450.
  • J. Rada and M. Saorin, Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra, 26(3) (1998), 899-912.
  • J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  • D. G. Wang, Modules with flat socle and almost excellent extensions, Acta Math. Vietnam, 21 (1996), 295-301.
  • R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc., 155 (1971), 233-256.
  • R. Wisbauer, Foundations of Module and Ring Theory, Philadelphia, Gordon and Breach, 1991.
  • Y. F. Xiao, Rings with flat socles, Proc. Amer. Math. Soc., 123 (1995), 2391-2395.
  • W. M. Xue, Modules with projective socle, Riv. Mat. Univ., 5 (1992), 311-315.
  • X. X. Zhang and J. L. Chen, A note on relative flatness and coherence, Comm. Algebra, 35 (2007), 3321-3330.