Do generating sets of ideals increase if ideals increase?

Question

Suppose I have a ring $R$ (let's say an integral domain), and two finitely generated ideals $I\subseteq J$. Is it true that the minimum size of a generating set of $I$ can't be larger than the minimum size of a generating set of $J$?

I can't prove it but I also can't think of a counterexample.

Answer 1

Let $J = R$ and let $I \subsetneq R$ be a finitely-generated non-principal ideal. Then $J = (1)$ but $(x) \subsetneq I$ for every $x \in I$. Take for example $R = \mathbb{Q}[X,Y]$ and $I = (X,Y)$.

Answer 2

I intended for $J\neq R$ but the principle of Jendrik's answer extends: Let $J=(a)$ be some principal ideal and $I_0$ be a non-principal ideal (say, $I_0=(x,y)$). Then define $I:=I_0J=(ax,ay)\subseteq J$.

If $I$ is not principal, that disproves the claim. If $I$ is principal, let's say $I=(z)$, then $ax=bz$ and $ay=cz$ for some $b,c\in R$. But also, $(z)=I\subseteq J=(a)$, so $z\in (a)$ and hence $z=da$ for some $d$. Thus $ax=bda$ and $ay=cda$, and thus $x=bd$ and $y=cd$. But this means $I_0=(x,y)\subseteq (d)$, which also disproves the claim.