Coherent rings of finite weak global dimension

Enochs, Edgar; Hernández, Juan; del Valle, Alberto

doi:10.1090/S0002-9939-98-04191-4

Coherent rings of finite weak global dimension
HTML articles powered by AMS MathViewer

by Edgar E. Enochs, Juan Martínez Hernández and Alberto del Valle PDF

Proc. Amer. Math. Soc. 126 (1998), 1611-1620 Request permission

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

J. Asensio Mayor and J. Martínez Hernández, Monomorphic flat envelopes in commutative rings, Arch. Math. (Basel) 54 (1990), no. 5, 430–435. MR 1049197, DOI 10.1007/BF01188669
J. Asensio Mayor and J. Martínez Hernández, On flat and projective envelopes, J. Algebra 160 (1993), no. 2, 434–440. MR 1244922, DOI 10.1006/jabr.1993.1195
Richard Belshoff, Edgar E. Enochs, and Jin Zhong Xu, The existence of flat covers, Proc. Amer. Math. Soc. 122 (1994), no. 4, 985–991. MR 1209416, DOI 10.1090/S0002-9939-1994-1209416-4
Ladislav Bican, T. Kepka, and P. Němec, Rings, modules, and preradicals, Lecture Notes in Pure and Applied Mathematics, vol. 75, Marcel Dekker, Inc., New York, 1982. MR 655412
Jose Luis Bueso Montero, Blas Torrecillas Jover, and Alain Verschoren, Local cohomology and localization, Pitman Research Notes in Mathematics Series, vol. 226, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1088249
Edgar E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), no. 3, 189–209. MR 636889, DOI 10.1007/BF02760849
Edgar Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179–184. MR 754698, DOI 10.1090/S0002-9939-1984-0754698-X
José L. Gómez Pardo and Nieves Rodríguez González, On some properties of IF rings, Quaestiones Math. 5 (1983), no. 4, 395–405. MR 700517
J. Martínez Hernández, M. Saorín, and A. del Valle, Noncommutative rings whose modules have essential flat envelopes, J. Algebra 177 (1995), no. 2, 434–450. MR 1355209, DOI 10.1006/jabr.1995.1306
B. L. Osofsky, Global dimension of valuation rings, Trans. Amer. Math. Soc. 127 (1967), 136–149. MR 206074, DOI 10.1090/S0002-9947-1967-0206074-0
Judith D. Sally and Wolmer V. Vasconcelos, Flat ideals I, Comm. Algebra 3 (1975), 531–543. MR 379466, DOI 10.1080/00927877508822059
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953
Wolmer V. Vasconcelos, The rings of dimension two, Lecture Notes in Pure and Applied Mathematics, Vol. 22, Marcel Dekker, Inc., New York-Basel, 1976. MR 0427290
Jin Zhong Xu, The existence of flat covers over Noetherian rings of finite Krull dimension, Proc. Amer. Math. Soc. 123 (1995), no. 1, 27–32. MR 1242111, DOI 10.1090/S0002-9939-1995-1242111-5