Let R be a local ring with maximal ideal J. Let M be a finitely generated R-module, and let V=M/JM. Then if \{x_1+JM,...,x_n+JM\} is a basis for V over R/J, then \{x_1, ... , x_n\} is a minimal set of generators for M.
Proof
Let N=\sum_{i=1}^n Rx_i. Since x_i + JM generate V=M/JM, we have...