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Test Modules for Projectivity

P. Jothilingam
Proceedings of the American Mathematical Society
Vol. 94, No. 4 (Aug., 1985), pp. 593-596
DOI: 10.2307/2044870
Stable URL: http://www.jstor.org/stable/2044870
Page Count: 4
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Test Modules for Projectivity
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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 $\operatorname{Ext}^1_R(P, M) = (0)$, 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 ⩽ 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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