ON n-SEMIHEREDITARY AND n-COHERENT RINGS

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

Xiaoxiang Zhang Jianlong Chen

https://izlik.org/JA27PN38JE

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

Keywords

(weakly) n-semihereditary ring , n-coherent ring

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: [e-mail] (X. Zhang), [e-mail] (J. Chen)