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Every module is an inverse limit of injectives
Author:
George M. Bergman
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1177-1183
MSC (2010):
Primary 16D50, 18A30; Secondary 13C11, 16D90
Posted:
August 28, 2012
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Additional Information
Abstract: It is shown that any left module  over a ring  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  is left Noetherian,  can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised.
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- 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.
- 2.
- George M. Bergman, Some empty inverse limits, unpublished note, Oct. 2005, revised July 2011. Readable at http://math.berkeley.edu/~gbergman/papers/unpub.
- 3.
- Hyman Bass, Injective dimension in Noetherian rings, Trans. Amer. Math. Soc. 102 (1962) 18-29. MR 0138644 (25:2087)
- 4.
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, 1956, reprinted 1999. MR 1731415 (2000h:18022)
- 5.
- Carl Faith, Algebra. II. Ring theory, Grundlehren der mathematischen Wissenschaften, No. 191. Springer, 1976. MR 0427349 (55:383)
- 6.
- V.E.Govorov, On flat modules (Russian), Sibirsk. Mat. Ž. 6 (1965) 300-304. MR 0174598 (30:4799)
- 7.
- Leon Henkin, A problem on inverse mapping systems, Proc. Amer. Math. Soc. 1 (1950) 224-225. MR 0035006 (11:675e)
- 8.
- Graham Higman and A.H.Stone, On inverse systems with trivial limits, J. London Math. Soc. 29 (1954) 233-236. MR 0061086 (15:773b)
- 9.
- Thomas W. Hungerford, Algebra, Holt, Rinehart and Winston, 1974; reprinted as Springer Graduate Texts in Math., 73, 1980. MR 600654 (82a:00006)
- 10.
- T. Y. Lam, Lectures on modules and rings, Springer Graduate Texts in Math., vol. 189, 1999. MR 1653294 (99i:16001)
- 11.
- Daniel Lazard, Sur les modules plats, C. R. Acad. Sci. Paris 258 (1964) 6313-6316. MR 0168625 (29:5883)
- 12.
- Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958) 511-528. MR 0099360 (20:5800)
- 13.
- Zoltán Papp, On algebraically closed modules, Publ. Math. Debrecen 6 (1959) 311-327. MR 0121390 (22:12128)
- 14.
- L. Salce, Classi di gruppi abeliani chiuse rispetto alle immagini omomorfe ed ai limiti proiettivi, Rend. Sem. Mat. Univ. Padova 49 (1973) 1-7. MR 0357653 (50:10121)
- 15.
- William C. Waterhouse, An empty inverse limit, Proc. Amer. Math. Soc. 36 (1972) 618. MR 0309047 (46:8158)
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Additional Information
George M. Bergman
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720-3840
Email:
gbergman@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11453-4
PII:
S 0002-9939(2012)11453-4
Keywords:
Inverse limit of injective modules
Received by editor(s):
April 15, 2011
Received by editor(s) in revised form:
August 16, 2011
Posted:
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
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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