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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Test modules for projectivity


Author: P. Jothilingam
Journal: Proc. Amer. Math. Soc. 94 (1985), 593-596
MSC: Primary 13C10; Secondary 13C15
DOI: https://doi.org/10.1090/S0002-9939-1985-0792267-7
MathSciNet review: 792267
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Abstract: Let $ R$ be a commutative noetherian local ring with identity. Modules over $ R$ will be assumed to be finitely generated and unitary. A nonzero $ R$-module $ M$ is said to be a strong test module for projectivity if the condition $ \operatorname{Ext}_R^1(P,M) = (0)$, for an arbitrary module $ P$, implies that $ P$ is projective. This definition is due to Mark Ramras [5]. He proves that a necessary condition for $ M$ to be a strong test module is that depth $ M \leqslant 1$. This is also easy to see. In this note it is proved that, over a regular local ring, this condition is also sufficient for $ M$ to qualify as a strong test module.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0792267-7
Keywords: Projective dimension, Ext, Tor, commutative noetherian local ring, depth, Krull dimension, artinian module
Article copyright: © Copyright 1985 American Mathematical Society