Injective endormorphisms of finitely generated modules
Author:
Wolmer V. Vasconcelos
Journal:
Proc. Amer. Math. Soc. 25 (1970), 900-901
MSC:
Primary 13.40
DOI:
https://doi.org/10.1090/S0002-9939-1970-0258814-8
MathSciNet review:
0258814
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $R$ be a commutative ring. Then any infective endomorphism of a finitely generated $R$-module is always an isomorphism if and only if $R$ is $0$-dimensional, that is, if every prime ideal is maximal.
- N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1261, Hermann, Paris, 1958 (French). MR 0098114
- A. Grothendieck, Groupes diagonalisables, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64) Inst. Hautes Études Sci., Paris, 1964, pp. 36 (French). MR 0212024
- Jan R. Strooker, Lifting projectives, Nagoya Math. J. 27 (1966), 747–751. MR 197514
- Wolmer V. Vasconcelos, On local and stable cancellation, An. Acad. Brasil. Ci. 37 (1965), 389–393. MR 214591
- Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI https://doi.org/10.1090/S0002-9947-1969-0238839-5
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Additional Information
Keywords:
Injective endomorphism,
finitely generated module,
Krull dimension zero
Article copyright:
© Copyright 1970
American Mathematical Society