TITLE: Finite modules over finite local rings
QUESTION [8 upvotes]: Let $R$ be a finite commutative local ring with identity. If $M$ is a finite $R$-module it is necessarily projective?

REPLY [6 votes]: Let $p$ be a prime number.
Let $R = \mathbb{Z}/p^2\mathbb{Z}$.
Let $M = R/pR$.
Since the number of elements of $M$ is $p$, $M$ cannot be free.
Hence $M$ cannot be projective.