∏-COHERENT DIMENSIONS AND ∏-COHERENT RINGS

Title & Authors
∏-COHERENT DIMENSIONS AND ∏-COHERENT RINGS
Mao, Lixin;

Abstract
R is called a right $\small{{\Pi}-coherent}$ ring in case every finitely generated torsion less right R-module is finitely presented. In this paper, we define a dimension for rings, called $\small{{\Pi}-coherent}$ dimension, which measures how far away a ring is from being $\small{{\Pi}-coherent}$. This dimension has nice properties when the ring in question is coherent. In addition, we study some properties of $\small{{\Pi}-coherent}$ rings in terms of preenvelopes and precovers.
Keywords
$\small{{\Pi}-coherent}$ dimension;$\small{{\Pi}-coherent}$ ring;FGT-injective module;FGT-flat module;FGT-injective dimension;preenvelope;precover;
Language
English
Cited by
1.
PRECOVERS AND PREENVELOPES BY MODULES OF FINITE FGT-INJECTIVE AND FGT-FLAT DIMENSIONS,;

대한수학회논문집, 2010. vol.25. 4, pp.497-510
1.
PRECOVERS AND PREENVELOPES BY MODULES OF FINITE FGT-INJECTIVE AND FGT-FLAT DIMENSIONS, Communications of the Korean Mathematical Society, 2010, 25, 4, 497
2.
FGT-injective dimensions of Π-coherent rings and almost excellent extension, Proceedings - Mathematical Sciences, 2010, 120, 2, 149
References
1.
L. Bican, R. EI Bashir, and E. E. Enochs, All modules have fiat covers, Bull. London Math. Soc. 33 (2001), 385-390

2.
L. Bonami, On the Structure of Skew Group Rings, Algebra Berichte 48, Verlag Reinhard Fisher, Munchen, 1984

3.
V. Camillo, Coherence for polynomial rings, J. Algebra 132 (1990), 72-76

4.
F. C. Chen, J. Y. Tang, Z. Y. Huang, and M. Y. Wang, \${\Pi}\$-coherent rings and FGTinjective dimension, Southeast Asian Bulletin Math. 19 (1995), no. 3, 105-112

5.
N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459-1470

6.
N. Q. Ding and J. L. Chen, Relative coherence and preenvelopes, Manuscripta Math; 81 (1993), 243-262

7.
P. C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), no. 1,41-51

8.
E. E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (1976), 361-362

9.
E. E. Enochs, Injective and fiat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189-209

10.
E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter, Berlin-New York, 2000

11.
S. Jain, Flat and F P-injectivity, Proc. Amer. Math. Soc. 41 (1973), 437-442

12.
M. F. Jones, Flatness and f-projectivity of torsion-free modules and injective modules, Lecture Notes in Math. 951 (1982), 94-116

13.
T. Y. Lam, Lectures on Modules and Rings; Springer-Verlag, New York-HeidelbergBerlin, 1999

14.
Z. K. Liu, Excellent extensions and homological dimensions, Comm. Algebra 22 (1994), no. 5, 1741-1745

15.
B. Madox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158

16.
L. X. Mao and N. Q. Ding, FP-projective dimensions, Comm. Algebra 33 (2005), no. 4, 1153-1170

17.
K. R. Pinzon, Absolutely pure modules, University of Kentucky, Ph. D thesis, 2005

18.
J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979

19.
B. Stenstrom, Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323-329

20.
J. Trlifaj, Covers, Envelopes, and Cotorsion Theories, Lecture notes for the workshop, 'Homological Methods in Module Theory'. Cortona, September 10-16, 2000

21.
M. Y. Wang, Some studies on \${\Pi}\$-coherent rings, Proc. Amer. Math. Soc. 119 (1993), 71-76

22.
J. Xu, Flat Covers of Modules, Lecture Notes in Math. 1634, Springer-Verlag: BerlinHeidelberg-New York, 1996