Some properties of AF G and CT F rings

Some properties of AF G and CT F rings

R is said to be a right AF G ring if the right annihilator of everynonempty subset of R is a finitely generated right ideal. R is calleda right CT F ring if every cyclic torsionless right R-module embeds in afree module. In this paper, we first give new characterizations of AF Grings and study some closure properties of AF G rings. Then we explorethe intimate relationships between AF G rings and CT F rings.

___

  • 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), 73.
  • V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), 72-76.
  • S.U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473.
  • T.J. Cheatham and D.R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc. 81 (1981), 175-177.
  • R.R. Colby, Rings which have flat injective modules, J. Algebra 35 (1975), 239-252.
  • E.E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (1976), 361-362.
  • E.E. Enochs and O.M.G. Jenda, Relative Homological Algebra; Walter de Gruyter: Berlin- New York, 2000.
  • C. Faith, Rings with ascending chain condition on annihilators, Nagoya Math. J. 27 (1966), 191.
  • C. Faith, Embedding torsionless modules in projectives, Publ. Mat. 34 (1990), 379-387.
  • S. Glaz, Commutative Coherent Rings; Lecture Notes in Math. 1371, Springer-Verlag: New York, 1989.
  • J.L.Gómez Pardo, Embedding cyclic and torsionfree modules in free modules, Arch. Math. (1985), 503-510.
  • M. Ikeda and T. Nakayama, On some characteristic properties of quasi-Frobenius and regular rings, Proc. Amer. Math. Soc. 5 (1954), 15-19.
  • F. Kasch, Modules and Rings; Academic Press: London-New York, 1982.
  • T.Y. Lam, Lectures on Modules and Rings; Springer-Verlag: New York-Heidelberg-Berlin, B. Madox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158.
  • L.X. Mao, Rings close to Baer, Indian J. Pure Appl. Math. 38 (2007), 129-142.
  • L.X. Mao, A generalization of Noetherian rings, Taiwanese J. Math. 12 (2008), 501-512.
  • L.X. Mao, Baer endomorphism rings and envelopes, J. Algebra Appl. 9 (2010), 365-381.
  • L.X. Mao, Properties of P -coherent and Baer modules, Period. Math. Hungar. 60 (2010), 114.
  • L.X. Mao and N.Q. Ding, New characterizations of pseudo-coherent rings, Forum Math. 22 (2010), 993-1008.
  • W.K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings; Cambridge Tracts in Math. 158, Cambridge University Press, 2003.
  • J.J. Rotman, An Introduction to Homological Algebra; Academic Press: New York, 1979.
  • B. Stenström, Coherent rings and F P -injective modules, J. London Math. Soc. 2 (1970), 329.
  • R. Wisbauer, Foundations of Module and Ring Theory; Gordon and Breach, 1991.
  • W. Xue, Rings related to quasi-Frobenius rings, Algebra Colloq. 4 (1998), 471-480.
  • H.Y. Zhu and N.Q. Ding, Generalized morphic rings and their applications, Comm. Algebra (2007), 2820-2837.