I have seen the phrase "minimal number of generators of an ideal" (in a Noetherian local ring) several times. I am unable to see how this is a well defined. Explicitly, how do we show, if $x_1,...,x_m$ and $y_1,...,y_n$ are minimal generating sets of an ideal in a Noetherian local ring, then $m=n$.
HINT $\ $ Over a local ring there is a nicely behaved notion of minimal set of generators by way of Nakayama's lemma. See section 20.1, The uniqueness of free resolutions, in Eisenbud: Commutative Algebra: with a View Toward Algebraic Geometry.