Taiwanese Journal of Mathematics


Xian-Hui Fu and Nan-Qing Ding

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Let $R$ be a ring with identity and let $\mathcal{M}_R$ be the category of right $R$-modules. In this article, we study some relations between torsion theories and cotorsion theories in $\mathcal{M}_R$. As applications, we give some new characterizations of IF rings with essential flat envelopes.

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Taiwanese J. Math., Volume 14, Number 4 (2010), 1249-1270.

First available in Project Euclid: 18 July 2017

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Primary: 16D40: Free, projective, and flat modules and ideals [See also 19A13] 16S90: Torsion theories; radicals on module categories [See also 13D30, 18E40] {For radicals of rings, see 16Nxx} 18E40: Torsion theories, radicals [See also 13D30, 16S90]

torsion theory cotorsion theory flat envelope IF ring


Fu, Xian-Hui; Ding, Nan-Qing. TORSION THEORIES AND ESSENTIAL FLAT ENVELOPES. Taiwanese J. Math. 14 (2010), no. 4, 1249--1270. doi:10.11650/twjm/1500405942. https://projecteuclid.org/euclid.twjm/1500405942

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  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
  • J. Asensio Mayor and J. Mart\'\tiny lnez Hernández, Flat envelopes in commutative rings, Israel J. Math. 62(1) (1988), 123-128.
  • J. Asensio Mayor and J. Mart\'\tiny lnez Hernández, Monomorphic flat envelopes in commutative rings, Arch. Math., 54 (1990), 430-435.
  • J. Asensio Mayor and J. Mart\'\tiny lnez Hernández, On flat and projective envelopes, J. Algebra 160 (1993), 434-440.
  • A. Beligiannis and I. Reiten, Homological and Homotopical Aspects of Torsion Theories, Mem. Amer. Math. Soc. 188. Providence, RI: American Mathematical Society, 2007.
  • L. Bican, R. El Bashir and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc., 33 (2001), 385-390. \def\vee$\check{\rm e}$
  • L. Bican, T. Kepka and P. N\vee mec, Rings, Modules and Preradicals, Marcel Dekker, New York, 1982.
  • T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81(2) (1981), 175-177.
  • J. L. Chen, On von Neumann regular rings and SF-rings, Math. Japonica, 36 (1991), 1123-1127.
  • R. R. Colby, Rings which have flat injective modules, J. Algebra, 35 (1975), 239-252.
  • S. E. Dickson, A torsion theory for abelian categories, Trans. Amer. Soc., 121 (1963), 223-235.
  • N. Q. Ding and J. L. Chen, Relative coherence and preenvelopes, Manus. Math., 81 (1993), 243-262.
  • E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math., 39 (1981), 189-209.
  • E. E. Enochs, J. Mart\'\tiny lnez Hernández and A. Del Valle, Coherent rings of finite weak global dimension, Proc. Amer. Math. Soc., 126(6) (1998), 1611-1620.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
  • R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, Walter de Gruyter, Berlin-New York, 2006.
  • J. L. Gómez Pardo and N. Rodr\'\tiny lguez González, On some properties of IF rings, Quaestiones Math., 5 (1983), 395-405.
  • K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules, Marcel Dekker, New York, 1976.
  • P. A. Guil Asensio and I. Herzog, Sigma-cotorsion rings, Adv. Math., 191 (2005), 11-28.
  • D. Lazard, Autor de la platitude, Bull. Soc. Math. France, 97 (1968), 81-128.
  • L. X. Mao and N. Q. Ding, Notes on cotorsion modules, Comm. Algebra, 33(1) (2005), 349-360.
  • L. X. Mao and N. Q. Ding, Cotorsion modules and relative pure-injectivity, J. Austral. Math. Soc., 81 (2006), 225-243.
  • L. X. Mao and N. Q. Ding, $\mathcal{L}$-injective hulls of modules, Bull. Austral. Math. Soc., 74 (2006), 37-44.
  • J. Mart\'\tiny lnez Hernández, M. Saor\'\tiny ln and A. Del Valle, Noncommutative rings whose modules have essential flat envelopes, J. Algebra 177 (1995), 434-450.
  • S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, Cambridge Univ. Press, Cambridge, UK, 1990.
  • K. Ohtake and H. Tachikawa, Colocalization and localization in abelian categories, J. Algebra, 56 (1979), 1-23.
  • V. S. Ramamurthi, On the injectivity and flatness of certain cyclic modules, Proc. Amer. Math. Soc., 48(1) (1975), 21-25.
  • J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  • M. Saor\'\tiny ln, The structure of commutative rings with monomorphic flat envelopes, Comm. Algebra, 23(14) (1995), 5383-5394.
  • L. Salce, Cotorsion theories for abelian groups, Symposia Math., XXIII (1979), 11-32.
  • B. Stenström, Rings of Quotients, Springer-Verlag, Berlin-Heidelberg-New York, 1975.
  • R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, 1991.
  • J. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag, Berlin-Heidelberg-New York, 1996.