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## ON n-SEMIHEREDITARY AND n-COHERENT RINGS

#### Xiaoxiang Zhang [1] , Jianlong Chen [2]

##### 95 158

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
(weakly) n-semihereditary ring, n-coherent ring
• 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)
Other ID JA66CV45JH Articles Author: Xiaoxiang Zhang Author: Jianlong Chen Publication Date: June 1, 2007
 Bibtex @ { ieja266407, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Abdullah HARMANCI}, year = {2007}, volume = {1}, pages = {1 - 10}, doi = {}, title = {ON n-SEMIHEREDITARY AND n-COHERENT RINGS}, key = {cite}, author = {Zhang, Xiaoxiang and Chen, Jianlong} } APA Zhang, X , Chen, J . (2007). ON n-SEMIHEREDITARY AND n-COHERENT RINGS. International Electronic Journal of Algebra, 1 (1), 1-10. Retrieved from http://dergipark.org.tr/ieja/issue/25210/266407 MLA Zhang, X , Chen, J . "ON n-SEMIHEREDITARY AND n-COHERENT RINGS". International Electronic Journal of Algebra 1 (2007): 1-10 Chicago Zhang, X , Chen, J . "ON n-SEMIHEREDITARY AND n-COHERENT RINGS". International Electronic Journal of Algebra 1 (2007): 1-10 RIS TY - JOUR T1 - ON n-SEMIHEREDITARY AND n-COHERENT RINGS AU - Xiaoxiang Zhang , Jianlong Chen Y1 - 2007 PY - 2007 N1 - DO - T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 1 EP - 10 VL - 1 IS - 1 SN - 1306-6048-1306-6048 M3 - UR - Y2 - 2019 ER - EndNote %0 International Electronic Journal of Algebra ON n-SEMIHEREDITARY AND n-COHERENT RINGS %A Xiaoxiang Zhang , Jianlong Chen %T ON n-SEMIHEREDITARY AND n-COHERENT RINGS %D 2007 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 1 %N 1 %R %U ISNAD Zhang, Xiaoxiang , Chen, Jianlong . "ON n-SEMIHEREDITARY AND n-COHERENT RINGS". International Electronic Journal of Algebra 1 / 1 (June 2007): 1-10. AMA Zhang X , Chen J . ON n-SEMIHEREDITARY AND n-COHERENT RINGS. IEJA. 2007; 1(1): 1-10. Vancouver Zhang X , Chen J . ON n-SEMIHEREDITARY AND n-COHERENT RINGS. International Electronic Journal of Algebra. 2007; 1(1): 10-1.