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n-X-COHERENT RINGS

Year 2010, Volume: 7 Issue: 7, 128 - 139, 01.06.2010

Abstract

This paper unifies several generalizations of coherent rings in one notion. Namely, we introduce n-X -coherent rings, where X is a class of modules and n is a positive integer, as those rings for which the subclass Xn of n-presented modules of X is not empty, and every module in Xn is n + 1-presented. Then, for each particular class X of modules, we find correspondent relative coherent rings. Our main aim is to show that the well-known Chase’s, Cheatham and Stone’s, Enochs’, and Stenstr¨om’s characterizations of coherent rings hold true for any n-X -coherent rings.

References

  • N. Bourbaki, Alg`ebre Commutative, Chapitre 1-2, Masson, Paris, 1961.
  • N. Bourbaki, Alg`ebre Homologique, Chapitre 10, Masson, Paris, 1980.
  • 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), –473.
  • T. J. Cheatham, D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81 (1981), 175–177.
  • D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Alge- bra, 22 (1994), 3997–4011.
  • J. Chen, N. Ding, On n-Coherent rings, Comm. Algebra, 24 (1996), 3211–216.
  • N. Ding, Y. Li, L. Mao, J-coherent rings, J. Algebra Appl., 8 (2009), 139–155.
  • N. Ding and L. Mao, On a new characterization of coherent rings, Publ. Math. (Debreen), 71 (2007), 67–82
  • N. Ding, L. Mao, On divisible and torsionfree modules, Comm. Algebra, 36 (2008), 708–731
  • D. E. Dobbs, S. Kabbaj, N. Mahdou, M. Sobrani, When is D + M n-coherent and an (n, d)-domain?, Lecture Notes in Pure and Appl. Math., Dekker, 205 (1999), 257–270.
  • D. E. Dobbs, S. Kabbaj, N. Mahdou, n-coherent rings and modules, Lecture Notes in Pure and Appl. Math., Dekker, 185 (1997), 269–281.
  • E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., Springer- Verlag, Berlin, 1989.
  • S. Kabbaj, N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra, 32 (2004), 3937–3953.
  • J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull., 7 (1964), 237–243
  • S. B. Lee, n-coherent rings, Comm. Algebra, 30 (2002), 1119–1126.
  • L. Li, J. Wei, Generalized coherent rings by Gorenstein projective dimension, Matematicki vesnik, 60 (2008), 155–163.
  • L. Mao, Π-Coherent dimension and Π-Coherent rings, J. Korean Math. Soc., (2007), 719–731.
  • L. Mao, Min-Flat Modules and Min-Coherent Rings, Comm. Algebra, 35 (2007), 635–650.
  • J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  • B. Stenstr¨om, Coherent rings and FP-injective modules, J. London Math. Soc., 2 (1970), 323–329.
  • B. Stenstr¨om, Rings of Quotients, Springer-Verlag: Berlin, Heidelberg, New York, 1975.
  • X. Zhang, J. Chen, J. Zhang, On (m, n)-injective modules and (m, n)-coherent rings, Algebra Colloq., 12 (2005), 149–160. Driss Bennis
  • Department of Mathematics Faculty of Science and Technology of Fez, Box 2202
  • University S. M. Ben Abdellah Fez, Morocco e-mail: driss bennis@hotmail.com
Year 2010, Volume: 7 Issue: 7, 128 - 139, 01.06.2010

Abstract

References

  • N. Bourbaki, Alg`ebre Commutative, Chapitre 1-2, Masson, Paris, 1961.
  • N. Bourbaki, Alg`ebre Homologique, Chapitre 10, Masson, Paris, 1980.
  • 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), –473.
  • T. J. Cheatham, D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81 (1981), 175–177.
  • D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Alge- bra, 22 (1994), 3997–4011.
  • J. Chen, N. Ding, On n-Coherent rings, Comm. Algebra, 24 (1996), 3211–216.
  • N. Ding, Y. Li, L. Mao, J-coherent rings, J. Algebra Appl., 8 (2009), 139–155.
  • N. Ding and L. Mao, On a new characterization of coherent rings, Publ. Math. (Debreen), 71 (2007), 67–82
  • N. Ding, L. Mao, On divisible and torsionfree modules, Comm. Algebra, 36 (2008), 708–731
  • D. E. Dobbs, S. Kabbaj, N. Mahdou, M. Sobrani, When is D + M n-coherent and an (n, d)-domain?, Lecture Notes in Pure and Appl. Math., Dekker, 205 (1999), 257–270.
  • D. E. Dobbs, S. Kabbaj, N. Mahdou, n-coherent rings and modules, Lecture Notes in Pure and Appl. Math., Dekker, 185 (1997), 269–281.
  • E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., Springer- Verlag, Berlin, 1989.
  • S. Kabbaj, N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra, 32 (2004), 3937–3953.
  • J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull., 7 (1964), 237–243
  • S. B. Lee, n-coherent rings, Comm. Algebra, 30 (2002), 1119–1126.
  • L. Li, J. Wei, Generalized coherent rings by Gorenstein projective dimension, Matematicki vesnik, 60 (2008), 155–163.
  • L. Mao, Π-Coherent dimension and Π-Coherent rings, J. Korean Math. Soc., (2007), 719–731.
  • L. Mao, Min-Flat Modules and Min-Coherent Rings, Comm. Algebra, 35 (2007), 635–650.
  • J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.
  • B. Stenstr¨om, Coherent rings and FP-injective modules, J. London Math. Soc., 2 (1970), 323–329.
  • B. Stenstr¨om, Rings of Quotients, Springer-Verlag: Berlin, Heidelberg, New York, 1975.
  • X. Zhang, J. Chen, J. Zhang, On (m, n)-injective modules and (m, n)-coherent rings, Algebra Colloq., 12 (2005), 149–160. Driss Bennis
  • Department of Mathematics Faculty of Science and Technology of Fez, Box 2202
  • University S. M. Ben Abdellah Fez, Morocco e-mail: driss bennis@hotmail.com
There are 26 citations in total.

Details

Other ID JA57VD95BJ
Journal Section Articles
Authors

Driss Bennis This is me

Publication Date June 1, 2010
Published in Issue Year 2010 Volume: 7 Issue: 7

Cite

APA Bennis, D. (2010). n-X-COHERENT RINGS. International Electronic Journal of Algebra, 7(7), 128-139.
AMA Bennis D. n-X-COHERENT RINGS. IEJA. June 2010;7(7):128-139.
Chicago Bennis, Driss. “N-X-COHERENT RINGS”. International Electronic Journal of Algebra 7, no. 7 (June 2010): 128-39.
EndNote Bennis D (June 1, 2010) n-X-COHERENT RINGS. International Electronic Journal of Algebra 7 7 128–139.
IEEE D. Bennis, “n-X-COHERENT RINGS”, IEJA, vol. 7, no. 7, pp. 128–139, 2010.
ISNAD Bennis, Driss. “N-X-COHERENT RINGS”. International Electronic Journal of Algebra 7/7 (June 2010), 128-139.
JAMA Bennis D. n-X-COHERENT RINGS. IEJA. 2010;7:128–139.
MLA Bennis, Driss. “N-X-COHERENT RINGS”. International Electronic Journal of Algebra, vol. 7, no. 7, 2010, pp. 128-39.
Vancouver Bennis D. n-X-COHERENT RINGS. IEJA. 2010;7(7):128-39.