Let $(R, m)$ be a local ring and $M$ a finitely generated projective $R$-module.
Let $n$ be the number of elements in a minimal generating set of $M$. Then by Nakayama's lemma $M/m M$ any minimal generating set of $M/ m M$ as an $R$-module must have at least $n$ elements.
It seems that this implies that $M/mM$ has dimension $n$ as a $R/mR$ module. Why is this?