Perfect torsion theories
HTML articles powered by AMS MathViewer
- by Paul E. Bland
- Proc. Amer. Math. Soc. 41 (1973), 349-355
- DOI: https://doi.org/10.1090/S0002-9939-1973-0376758-0
- PDF | Request permission
Abstract:
The purpose of this paper is to introduce the study of perfect torsion theories on $\operatorname {Mod} R$ and to dualize the concept of divisible module. A torsion theory on $\operatorname {Mod} R$ is called perfect if every torsion module has a projective cover. It is shown for such a theory that the class of torsion modules is closed under projective covers if and only if the class of torsion free modules is closed under factor modules. In addition, it is shown that this condition on a perfect torsion theory is equivalent to its idempotent radical being an epiradical. Codivisible covers of modules are also introduced and we are able to show that any module which has a projective cover has a codivisible cover. Codivisible covers are then characterized in terms of the projective cover of the module and the torsion submodule of the kernel of the minimal epimorphism.References
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- Spencer E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235. MR 191935, DOI 10.1090/S0002-9947-1966-0191935-0
- B. Eckmann and A. Schopf, Über injektive Moduln, Arch. Math. (Basel) 4 (1953), 75–78 (German). MR 55978, DOI 10.1007/BF01899665
- J. P. Jans, Some aspects of torsion, Pacific J. Math. 15 (1965), 1249–1259. MR 191936
- Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459
- Bodo Pareigis, Radikale und kleine Moduln, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 1965 (1966), no. Abt. II, 185–199 (1966) (German). MR 204493
- Bo Stenström, Rings and modules of quotients, Lecture Notes in Mathematics, Vol. 237, Springer-Verlag, Berlin-New York, 1971. MR 0325663
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 349-355
- MSC: Primary 16A50
- DOI: https://doi.org/10.1090/S0002-9939-1973-0376758-0
- MathSciNet review: 0376758