let $(R,m)$ be a local ring and let $I,J$ be an ideals of $R$, let $\mu(I)$ be the minimal number of generators of $I$. I've proved that $\mu(IJ)\leq\mu(I)$ but somehow this claim does not feel right to me.
actually in here the authors go to great length to prove that if $\mu(I^n)\leq n$, then for every $r>n$, $\mu(I^r) \leq n$. while by my proof this is trivial.
my proof is:
$I/Im$ is vector space over $R/m$ and $(IJ+Im)/Im$ is subspace of $I/Im$ so the dimension of $(IJ+Im)/Im$ is less then or equals $\mu(I)$ so $IJ+Im$ can be generated by $\mu(I)$ elements. let $A$ be a set containing $\mu(I)$ generators of $IJ+Im$ so $(A)=IJ+Im$ and by Nakayama's lemma $(A)=IJ$.
is this right? and if not, where is my mistake?