Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Coherent rings of finite weak global dimension


Authors: Edgar E. Enochs, Juan Martínez Hernández and Alberto del Valle
Journal: Proc. Amer. Math. Soc. 126 (1998), 1611-1620
MSC (1991): Primary 13C11, 13D05, 16D40, 16E70
DOI: https://doi.org/10.1090/S0002-9939-98-04191-4
MathSciNet review: 1443151
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs, Xu) and, if the weak global dimension is at most two, every left module has a flat envelope (Asensio, Martínez). We will exploit these features of this category to study its objects. In particular, we will consider orthogonal complements (relative to the extension functor) of several classes of modules in this category. In the case of a commutative ring we describe an idempotent radical on its category of modules which, when the weak global dimension does not exceed 2, can be used to analyze the structure of the flat envelopes and of the ring itself.


References [Enhancements On Off] (What's this?)

  • 1. J. Asensio Mayor and J. Martínez Hernández, Monomorphic flat envelopes in commutative rings, Arch. Math. 54 (1990) 430-435. MR 91k:16002
  • 2. J. Asensio Mayor and J. Martínez Hernández, On flat and projective envelopes, J. Algebra 69 (1993) 434-440. MR 94k:16005
  • 3. R. Belshoff, E.E. Enochs and J. Xu, The existence of flat covers, Proc. Amer. Math. Soc. 122 (1994) 985-991. MR 95b:16001
  • 4. L. Bican, T. Kepka, and P. N\v{e}mec, Rings, Modules and Preradicals, Lecture Notes in Pure and Applied Math. 75, Marcel Dekker Inc., New York, 1982. MR 83e:16026
  • 5. J.L. Bueso, B. Torrecillas and A. Verschoren, Local Cohomology and Localization, Pitman Research Notes in Mathematical Series 226, Longman Scientific and Technical, Harlow, 1989. MR 93j:13022
  • 6. E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. series 39 (1981) 189-209. MR 83a:16031
  • 7. E.E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984) 179-184. MR 85j:13016
  • 8. J.L. Gómez Pardo and N. Rodríguez González, On some properties of IF rings, Quaestiones Math. series 5 (1983) 395-405. MR 85a:16015
  • 9. J. Martínez Hernández, M. Saorín and A. del Valle, Noncommutative rings whose modules have essential flat envelopes, J. Algebra 177 (1995) 434-450. MR 96j:16003
  • 10. B.L. Osofsky, Global dimension of valuation rings, Trans. Amer. Math.Soc. 127 (1967) 136-149. MR 34:5899
  • 11. J.D. Sally and W.V. Vasconcelos, Flat ideals I, Comm. Algebra 3 (1975) 531-543. MR 52:371
  • 12. B. Stenström, An introduction to methods of ring theory, Grundlehren der mathematischen Wissenchaften, 217, Springer-Verlag, Berlin, 1975. MR 52:10782
  • 13. W.V. Vasconcelos, The Rings of Dimension Two, Lecture Notes in Pure and Applied Math. 22, Marcel Dekker Inc., New York, 1976. MR 55:324
  • 14. J. Xu, The existence of flat covers over Noetherian rings of finite Krull dimension, Proc. Amer. Math. Soc. 123 (1995) 27-32. MR 95c:16004

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13C11, 13D05, 16D40, 16E70

Retrieve articles in all journals with MSC (1991): 13C11, 13D05, 16D40, 16E70


Additional Information

Edgar E. Enochs
Affiliation: (Edgar E. Enochs) Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email: enochs@ms.uky.edu

Juan Martínez Hernández
Email: juan@fcu.um.es

Alberto del Valle
Affiliation: (Juan Martínez Hernández and Alberto del Valle) Departamento de Matemáticas, Universidad de Murcia, 30001 Murcia, Spain
Email: alberto@fcu.um.es

DOI: https://doi.org/10.1090/S0002-9939-98-04191-4
Keywords: Coherent ring, weak global dimension, flat envelope
Received by editor(s): February 1, 1996
Received by editor(s) in revised form: November 19, 1996
Additional Notes: The second and third authors are supported by the DGICYT of Spain (PB93-0515-C02-02) and by the Comunidad Autónoma de la Región de Murcia (PIB94/25).
Dedicated: Dedicated to the memory of Professor Maurice Auslander
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society