Every module is an inverse limit of injectives
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- by George M. Bergman PDF
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Abstract:
It is shown that any left module $A$ over a ring $R$ can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If $R$ is left Noetherian, $A$ can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised.References
- George M. Bergman, Final Examination, Math 250B, U.C. Berkeley, 15 May, 1990. Readable at http://math.berkeley.edu/~gbergman/grad.hndts/90Sp_250B_FE.pdf .
- George M. Bergman, Some empty inverse limits, unpublished note, Oct. 2005, revised July 2011. Readable at http://math.berkeley.edu/~gbergman/papers/unpub .
- Hyman Bass, Injective dimension in Noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18–29. MR 138644, DOI 10.1090/S0002-9947-1962-0138644-8
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415
- Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349, DOI 10.1007/978-3-642-65321-6
- V. E. Govorov, On flat modules, Sibirsk. Mat. Ž. 6 (1965), 300–304 (Russian). MR 0174598
- Leon Henkin, A problem on inverse mapping systems, Proc. Amer. Math. Soc. 1 (1950), 224–225. MR 35006, DOI 10.1090/S0002-9939-1950-0035006-6
- Graham Higman and A. H. Stone, On inverse systems with trivial limits, J. London Math. Soc. 29 (1954), 233–236. MR 61086, DOI 10.1112/jlms/s1-29.2.233
- Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original. MR 600654, DOI 10.1007/978-1-4612-6101-8
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
- Daniel Lazard, Sur les modules plats, C. R. Acad. Sci. Paris 258 (1964), 6313–6316 (French). MR 168625
- Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360, DOI 10.2140/pjm.1958.8.511
- Zoltán Papp, On algebraically closed modules, Publ. Math. Debrecen 6 (1959), 311–327. MR 121390
- L. Salce, Classi di gruppi abeliani chiuse rispetto alle immagini omomorfe ed ai limiti proiettivi, Rend. Sem. Mat. Univ. Padova 49 (1973), 1–7 (Italian). MR 357653
- William C. Waterhouse, An empty inverse limit, Proc. Amer. Math. Soc. 36 (1972), 618. MR 309047, DOI 10.1090/S0002-9939-1972-0309047-X
Additional Information
- George M. Bergman
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- Email: gbergman@math.berkeley.edu
- Received by editor(s): April 15, 2011
- Received by editor(s) in revised form: August 16, 2011
- Published electronically: August 28, 2012
- Additional Notes: http://arXiv.org/abs/arXiv:1104.3173 . After publication of this note, updates, errata, related references, etc., if found, will be recorded at http://math.berkeley.edu/~gbergman/papers/.
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1177-1183
- MSC (2010): Primary 16D50, 18A30; Secondary 13C11, 16D90
- DOI: https://doi.org/10.1090/S0002-9939-2012-11453-4
- MathSciNet review: 3008865