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Remarks on Op and Towber Rings

Published online by Cambridge University Press:  20 November 2018

David Lissner  and
Anthony Geramita
David Lissner
Affiliation:
Syracuse University, Syracuse, New York
Anthony Geramita
Affiliation:
Queen's University, Kingston, Ontario
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In this paper all rings considered have identity and are commutative, and all modules are finitely generated. We shall make liberal use of the definitions and notation established in [6; 7].

Towber observed in [9] that a local Outer Product ring (OP-ring) must have v-dimension ≦ 2, and so a local OP-ring is either regular of global dimension ≦ 2 or it has infinite global dimension. Since the global dimension of a noetherian ring is the supremum of the global dimensions of its localizations, we immediately obtain the following result.

THEOREM 1.1. The global dimension of a noetherian OP-ring is either∞ or ≦ 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1109 - 1117
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Auslander, M. and Buchsbaum, D., Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957), 390–405.Google Scholar
2. Bass, H., K-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 5–60.Google Scholar
3. Bourbaki, N., Éléments de mathématique, Fasc. XXXI, Algèbre commutative, chapitre 7: Diviseurs, Actualités Sci. Indust., No. 1314 (Hermann, Paris, 1965).Google Scholar
4. Cartan, H. and Eilenberg, S., Homological algebra (Princeton Univ. Press, Princeton, N. J., 1956).Google Scholar
5. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, Vol. III (Cambridge Univ. Press, Cambridge, 1954).Google Scholar
6. Lissner, D., Outer product rings, Trans. Amer. Math. Soc. 116 (1965), 526–535.Google Scholar
7. Lissner, D. and Geramita, A., Towber rings, J. Algebra 15 (1970), 13–40.Google Scholar
8. J.-P., Serre, Modules projectifs et espaces fibres à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, 18 pp. (Secrétariat Mathématique, Paris, 1958).Google Scholar
9. Towber, J., Complete reducibility in exterior algebras over free modules, J. Algebra 10 (1968), 299-309. 1Q# Local rings with the outer product property, Illinois J. Math. 14 (1970), 194–197.Google Scholar
11. Zariski, O. and Samuel, P., Commutative algebra, Vol. I, The University Series in Higher Mathematics (Van Nostrand, Princeton, N. J., 1958).Google Scholar