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ON n-SEMIHEREDITARY AND n-COHERENT RINGS

Year 2007, Volume: 1 Issue: 1, 1 - 10, 01.06.2007

Abstract

Let R be a ring. For a fixed positive integer n, R is said to
be left n-semihereditary in case every n-generated left ideal is projective. R
is said to be weakly n-semihereditary if each n-generated left (and/or right)
ideal is flat. Some properties of n-semihereditary rings, respectively, weakly
n-semihereditary rings and n-coherent rings are investigated. It is also proved
that R is left n-semihereditary if and only if it is left n-coherent and weakly
n-semihereditary, if and only if the ring of n × n matrices over R is left 1-
semihereditary if and only if the class of all n-flat right R-modules form the
torsion-free class of a torsion theory. Some known results are extended or
obtained as corollaries

References

  • H. Al-Ezeh, M. A. Natsheh and D. Hussein, Some properties of the ring of continuous functions, Arch. Math., 51 (1988), 60-64.
  • D. D. Anderson and D. E. Dobbs, Flatness, LCM-stability, and related module- theoretic properties, J. Algebra, 112 (1988), 139-150.
  • S. U. Chase, Direct Products of Modules, Trans. Amer. Math. Soc., 97 (1960), 457-473.
  • S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J., 18 (1961), 13-25.
  • J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24 (1996), 3211-3216. [6] J. L. Chen and Y. Q. Zhou, Extentions of injectivity and coherent rings, Comm. Algebra, 34 (2006), 275-288.
  • D. L. Costa, Parameterizing families of non-notherian rings, Comm. Algebra, 22 (1994), 3997-4011.
  • J. Dauns and L. Fuchs, Torsion-freeness in rings with zero-divisors, J. Algebra Appl., 3 (2004), 221-237.
  • L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, Heidelberg, Berlin, 1976.
  • S. Jİndrup, p.p. rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28 (1971), 431-435.
  • T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1998. [12] C. W. Neville, Flat C(X)-modules and F spaces, Math. Proc. Cambridge Phi- los. Soc., 106 (1989), 237-244.
  • C. W. Neville, When is C(X) a coherent ring ? Proc. Amer. Math. Soc., 110 (1990), 505-508.
  • K. Samei, Flat submodules of free modules over commutative Bezout rings, Bull. Austral. Math. Soc., 71 (2005), 113-119.
  • A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra, 29 (2001), 2039-2050. [16] X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)- coherent rings, Algebra Colloq., 12 (2005), 149-160.
  • Z. M. Zhu, J. L. Chen and X. X. Zhang, On (m, n)-purity of modules, East-west J. Math., 5 (2003), 35-44.
  • Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Scientiae Mathematicae Japonicae, 62 (2005), 455-459.
  • Xiaoxiang Zhang and Jianlong Chen
  • Department of Mathematics, Southeast University
  • Nanjing 210096, P. R. China
  • e-mail: z990303@seu.edu.cn (X. Zhang), jlchen@seu.edu.cn (J. Chen)
Year 2007, Volume: 1 Issue: 1, 1 - 10, 01.06.2007

Abstract

References

  • H. Al-Ezeh, M. A. Natsheh and D. Hussein, Some properties of the ring of continuous functions, Arch. Math., 51 (1988), 60-64.
  • D. D. Anderson and D. E. Dobbs, Flatness, LCM-stability, and related module- theoretic properties, J. Algebra, 112 (1988), 139-150.
  • S. U. Chase, Direct Products of Modules, Trans. Amer. Math. Soc., 97 (1960), 457-473.
  • S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J., 18 (1961), 13-25.
  • J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24 (1996), 3211-3216. [6] J. L. Chen and Y. Q. Zhou, Extentions of injectivity and coherent rings, Comm. Algebra, 34 (2006), 275-288.
  • D. L. Costa, Parameterizing families of non-notherian rings, Comm. Algebra, 22 (1994), 3997-4011.
  • J. Dauns and L. Fuchs, Torsion-freeness in rings with zero-divisors, J. Algebra Appl., 3 (2004), 221-237.
  • L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New York, Heidelberg, Berlin, 1976.
  • S. Jİndrup, p.p. rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28 (1971), 431-435.
  • T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1998. [12] C. W. Neville, Flat C(X)-modules and F spaces, Math. Proc. Cambridge Phi- los. Soc., 106 (1989), 237-244.
  • C. W. Neville, When is C(X) a coherent ring ? Proc. Amer. Math. Soc., 110 (1990), 505-508.
  • K. Samei, Flat submodules of free modules over commutative Bezout rings, Bull. Austral. Math. Soc., 71 (2005), 113-119.
  • A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra, 29 (2001), 2039-2050. [16] X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)- coherent rings, Algebra Colloq., 12 (2005), 149-160.
  • Z. M. Zhu, J. L. Chen and X. X. Zhang, On (m, n)-purity of modules, East-west J. Math., 5 (2003), 35-44.
  • Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Scientiae Mathematicae Japonicae, 62 (2005), 455-459.
  • Xiaoxiang Zhang and Jianlong Chen
  • Department of Mathematics, Southeast University
  • Nanjing 210096, P. R. China
  • e-mail: z990303@seu.edu.cn (X. Zhang), jlchen@seu.edu.cn (J. Chen)
There are 19 citations in total.

Details

Other ID JA66CV45JH
Journal Section Articles
Authors

Xiaoxiang Zhang This is me

Jianlong Chen This is me

Publication Date June 1, 2007
Published in Issue Year 2007 Volume: 1 Issue: 1

Cite

APA Zhang, X., & Chen, J. (2007). ON n-SEMIHEREDITARY AND n-COHERENT RINGS. International Electronic Journal of Algebra, 1(1), 1-10.
AMA Zhang X, Chen J. ON n-SEMIHEREDITARY AND n-COHERENT RINGS. IEJA. June 2007;1(1):1-10.
Chicago Zhang, Xiaoxiang, and Jianlong Chen. “ON N-SEMIHEREDITARY AND N-COHERENT RINGS”. International Electronic Journal of Algebra 1, no. 1 (June 2007): 1-10.
EndNote Zhang X, Chen J (June 1, 2007) ON n-SEMIHEREDITARY AND n-COHERENT RINGS. International Electronic Journal of Algebra 1 1 1–10.
IEEE X. Zhang and J. Chen, “ON n-SEMIHEREDITARY AND n-COHERENT RINGS”, IEJA, vol. 1, no. 1, pp. 1–10, 2007.
ISNAD Zhang, Xiaoxiang - Chen, Jianlong. “ON N-SEMIHEREDITARY AND N-COHERENT RINGS”. International Electronic Journal of Algebra 1/1 (June 2007), 1-10.
JAMA Zhang X, Chen J. ON n-SEMIHEREDITARY AND n-COHERENT RINGS. IEJA. 2007;1:1–10.
MLA Zhang, Xiaoxiang and Jianlong Chen. “ON N-SEMIHEREDITARY AND N-COHERENT RINGS”. International Electronic Journal of Algebra, vol. 1, no. 1, 2007, pp. 1-10.
Vancouver Zhang X, Chen J. ON n-SEMIHEREDITARY AND n-COHERENT RINGS. IEJA. 2007;1(1):1-10.