Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T15:20:29.305Z Has data issue: false hasContentIssue false
  • English
  • Français

The G-Function of Macrae

Published online by Cambridge University Press:  20 November 2018

David E. Rush
David E. Rush*
Affiliation:
University of California, Riverside, Riverside, California
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.

Let R be a commutative ring with identity. A finitely generated R-module M is called a torsion module if the annihilator of M contains a non zero-divisor. In [18] MacRae proved the following

Theorem. If R is a noetherian ring then there is a map G with the following properties from the class of torsion R-modules of finite homological dimension to the set of integral invertible ideals of R.

(i) If M is a finitely generated torsion R-module with homological dimension ≦ 1 then G(M) = F(M), the first Fitting ideal of M.

(ii) If S is a multiplicative subset of R then G(Ms) = G(M) s.

(iii) If 0 → L → M → N → 0 is an exact sequence of torsion modules of finite homological dimension then G(M) = G(L)G(N).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 854 - 865
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bass, H., Algebraic K-iheory (W. A. Benjamin Inc., New York, 1968).Google Scholar
2. Bertin, J., Anneaux cohérents regulars, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), 1–2.Google Scholar
3. Bourbaki, N., Algebra commutative, Chapters 1 and 2 (Hermann, Paris, 1961).Google Scholar
4. Buchsbaum, D., Complexes associated with the minors of a matrix, Symposia Math. IV (1970), 255–283.Google Scholar
5. Buchsbaum, D. and Eisenbud, D., Lifting modules and a theorem on Finite Free Resolutions (to appear).Google Scholar
6. Burch, L., A note on the homology of ideals generated by three elements in local rings, Proc. Camb. Philos. Soc. 64 (1968), 949–952.Google Scholar
7. Burch, L., On ideals of finite homologuai dimension in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941–948.Google Scholar
8. Endo, S., Note on p.p. rings, Nagoya Math. J. 17 (1960), 167–170.Google Scholar
9. Endo, S., On semi-hereditary rings, J. Math. Soc. Japan 18 (1961), 109–119.Google Scholar
10. Hilbert, D., Uber die Théorie der Algebraischen Formen, Math. Ann. 86 (1890), 473–534 or Ges. Abh. Vol. 2, 199-257.Google Scholar
11. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, Mass., 1970).Google Scholar
12. Kaplansky, I., Commutative rings, Lecture Notes, Queen Mary College, London, 1966.Google Scholar
13. Kohn, P., Ideals generated by three elements, Proc. Amer. Math. Soc. 85 (1972), 55–58.Google Scholar
14. Kramer, H., Eine Bemerkung zu einer Vermutung von Lipman, Ark. Math. 20 (1969), 30–35.Google Scholar
15. Kramer, H., Einige Anwendungen der G-Function von MacRae, Ark. Math. 22 (1972), 479–490.Google Scholar
16. Lazard, D., Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.Google Scholar
17. Lipman, J., On the Jacobian ideal of the module of differentials, Proc. Amer. Math. Soc. 21 (1969), 422–426.Google Scholar
18. MacRae, R., On an application of the Fitting invariants, J. Algebra 2 (1965), 153–159.Google Scholar
19. Marot, J., Une generalization de la notion d'anneaux de valuation, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), 1451–1454.Google Scholar
20. Moen, S., Derivation modules and a criterion for regularity (to appear).Google Scholar
21. Ohm, J., Homological dimension and Euler maps (to appear in the proceedings of the Commutative ring theory conference, University of Kansas, 1972).Google Scholar
22. Quentel, Y., Sur la compacité du spectre minimal d'un anneaux, Bull. Soc. Math. France 99 (1971), 265–272.Google Scholar
23. Quentel, Y., Sur la théorème a“Auslander-Buchsbaum, C.R. Acad. Sci. Paris Sér. A-B 278 (1971), 880–881.Google Scholar
24. Salmon, P., Sulla fattorialita dette algèbre graduate e degli anelli locali, Rend. Sem. Mat. Univ. PadovaXLI (1968), 119-137.Google Scholar
25. Samuel, P., On unique factorization domains (Tata Institute of Fundamental Research, Bombay, 1964).Google Scholar
26. Small, L., Change of rings theorem, Proc. Amer. Math. Soc. 19 (1968), 662–668.Google Scholar
27. Vasconcelos, W., Annihilators of modules with a finite free resolution, Proc. Amer. Math. Soc. 29 (1971), 440–442.Google Scholar
28. Zariski, O. and Samuel, P., Commutative algebra (van Nostrand, New York, vol I (1968) vol II (1961)).Google Scholar