Taiwanese Journal of Mathematics


Lixin Mao

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In this paper, we introduce the concept of $AFG$ rings. $R$ is said to be a left $AFG$ ring in case the left annihilator of every nonempty subset of $R$ is a finitely generated left ideal. Some characterizations of $AFG$ rings and applications are obtained.

Article information

Taiwanese J. Math., Volume 12, Number 2 (2008), 501-512.

First available in Project Euclid: 20 July 2017

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Zentralblatt MATH identifier

Primary: 16D40: Free, projective, and flat modules and ideals [See also 19A13] 16L60: Quasi-Frobenius rings [See also 16D50] 16D10: General module theory

$AFG$ ring $CF$ ring $PP$ ring singly projective module preenvelope


Mao, Lixin. A GENERALIZATION OF NOETHERIAN RINGS. Taiwanese J. Math. 12 (2008), no. 2, 501--512. doi:10.11650/twjm/1500574170. https://projecteuclid.org/euclid.twjm/1500574170

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  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
  • G. Azumaya, Finite splitness and finite projectivity, J. Algebra, 106 (1987), 114-134.
  • J. E. Björk, Rings satisfying certain chain conditions, J. Reine Angew Math., 245 (1970), 63-73.
  • S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960), 457-473.
  • J. L. Chen and N. Q. Ding, A note on existence of envelopes and covers, Bull. Austral. Math. Soc., 54 (1996), 383-390.
  • 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.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math. 1371, Springer-Verlag, New York, 1989.
  • C. R. Hajarnavis, On dual rings and their modules, J. Algebra, 93 (1985), 253-266.
  • C. Huh, H. K. Kim and Y. Lee, $P.P.$ rings and generalized $P.P.$ rings, J. Pure Appl. Algebra, 167 (2002), 37-52.
  • M. Ikeda and T. Nakayama, On some characteristic properties of quasi-Frobenius and regular rings, Proc. Amer. Math. Soc., 5 (1954), 5-18.
  • S. Jain, Flat and $FP$-injectivity, Proc. Amer. Math. Soc., 41 (1973), 437-442.
  • J. Jøndrup, $p.p.$ rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28 (1971), 431-435.
  • I. Kaplansky, Rings of Operators, New York, Benjamin, 1965.
  • T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York-Heidelberg-Berlin, 1999.
  • L. X. Mao and N. Q. Ding, On relative injective modules and relative coherent rings, Comm. Algebra, 34(7) (2006), 2531-2545.
  • B. Stenström, Coherent rings and $FP$-injective modules, J. London Math. Soc., 2 (1970), 323-329.
  • 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.
  • W.M. Xue, On $PP$ rings, Kobe J. Math., 7 (1990), 77-80.