A property of torsion-free modules over left Ore domains
Author:
Arthur Van de Water
Journal:
Proc. Amer. Math. Soc. 25 (1970), 199-201
MSC:
Primary 16.90
DOI:
https://doi.org/10.1090/S0002-9939-1970-0257155-2
MathSciNet review:
0257155
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Abstract: It is well known that for an integral domain $A$, the property that a module is divisible if and only if it is injective is equivalent to the property that $A$ is a Dedekind domain. In this paper, it is shown that if $A$ is a left Ore domain, then a torsion-free left $A$-module is divisible if and only if it is injective.
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N. Bourbaki, Algèbre, Chapitre 1, Hermann, Paris, 1958.
- N. Bourbaki, Éléments de mathématique. Première partie. Fascicule VI. Livre II: Algèbre. Chapitre 2: Algèbre linéaire, Actualités Sci. Indust., No. 1236. Hermann, Paris, 1962 (French). Troisième édition, entièrement refondue. MR 0155831
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963), 132–151. MR 142586, DOI https://doi.org/10.4153/CJM-1963-016-1
- Oystein Ore, Linear equations in non-commutative fields, Ann. of Math. (2) 32 (1931), no. 3, 463–477. MR 1503010, DOI https://doi.org/10.2307/1968245
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Additional Information
Keywords:
Torsion-free modules,
injective modules,
injective hull,
divisible modules,
noncommutative rings,
left quotient field,
left Ore domain
Article copyright:
© Copyright 1970
American Mathematical Society