Test Modules for Projectivity

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.