## Solution: Kaplansky’s Commutative Rings 4.1.2

Problem. Let $R$ be a (commutative!) Noetherian local ring, $M\subset R$ its maximal ideal, and $A$ a finitely generated $R$-module. If ${\rm Ext}^1(A,R/M) = 0$ then $A$ is a free $R$-module. This problem will be a stepping stone to showing that a Noetherian local ring is regular if and only if the injective dimension of […]