Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-16T13:50:40.382Z Has data issue: false hasContentIssue false
  • English
  • Français

On the idempotence and stability of kernel functors

Published online by Cambridge University Press:  18 May 2009

Mark L. Teply
Mark L. Teply
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee WI 53201, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category R-mod of left R-modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, σ satisfies σ(M / σ(M)) = 0 for every M ∊ R-mod. To every kernel functor / there corresponds a unique topologizing filter ℒσ = {I Ⅰ σ (R/I) = R/I} of left ideals and a unique class ℱσ = {M ∊ R-mod Ⅰ σ(M) = M} that is closed under homomorphic images, submodules, and direct sums. The idempotence of σ is characterized by either of the following additional conditions:

(1) if I ∊ ℒσ, K ⊆ I, and (K:x) = {r ∊ R ∣ rx ∊ K} ∊ ℒσ for each x ∊ I, then K ∊ ℒ or

(2) ℱσ is closed under extensions of one member of ℱσ by another member of ℱσ Idempotent kernel functors are important since they are the tool used to construct localization functors. For M∊ R-mod, let E(M) denote the injective hull of M. A kernel functor σ is called stable if Mℱ implies that E(M) ∊ ℱσ For more information about kernel functors, see [6], [7], [14], and [15].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 1 , January 1995 , pp. 37 - 43
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

1.Bican, L., Jambor, P., Kepka, T. and Němec, P., Stable and costable preradicals, Ada Univ. Carolinae—Math. et Phys. 16 (1975), 63–69.Google Scholar
2.Byrd, K. A., Rings whose quasi-injective modules are injective, Proc. Amer. Math. Soc. 33 (1972), 235–240.CrossRefGoogle Scholar
3.Damiano, R. F. and Papp, Z., On consequences of stability, Comm. Algebra 9 (1981), 747–764.CrossRefGoogle Scholar
4.Fenrick, M. H., Conditions under which all preradical classes are hereditary torsion classes, Comm. Algebra 2 (1974), 365–376.CrossRefGoogle Scholar
5.Golan, J. S., Torsion theories, Pitman Monographs and Surveys in Pure and Applied Mathematics 29 (Longman Scientific and Technical, 1986).Google Scholar
6.Golan, J. S., Linear topologies on a ring: an overview, Pitman Research Notes in Mathematics Series 159 (Longman Scientific and Technical, 1987).Google Scholar
7.Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 10–47.CrossRefGoogle Scholar
8.Gordon, R. and Robson, J. C., Krull dimension, Mem. Amer. Math. Soc. 133 (American Mathematical Society, 1973).Google Scholar
9.Gordon, R. and Robson, J. C., The Gabriel dimension of a module, J. Algebra 29 (1974) 459–473.CrossRefGoogle Scholar
10.Handelman, D., Strongly semiprime rings, Pacific J. Math. 60 (1975), 115–122.CrossRefGoogle Scholar
11.Papp, Z., On stable noetherian rings, Trans. Amer. Math. Soc. 213 (1975), 107–114.CrossRefGoogle Scholar
12.Viola-Prioli, A. M. D. and Viola-Prioli, J. E., Rings whose kernel functors are linearly ordered, Pacific J. Math. 132 (1988), 21–34.CrossRefGoogle Scholar
13.Viola-Prioli, A. M. D. and Viola-Prioli, J. E., Asymmetry in the lattice of kernel functors, Glasgow Math. J. 33 (1991), 95–97.CrossRefGoogle Scholar
14.Viola-Prioli, A. M. D. and Viola-Prioli, J. E., Rings arising from conditions on preradicals, Proceedings of the XXIst Ohio State/Denison Conference (World Scientific Publishing, 1993), 343–349.Google Scholar
15.Viola-Prioli, J. E., When is every kernel functor idempotent?, Canad. J. Math. 27 (1975), 545–554.CrossRefGoogle Scholar