Example of $\mu(I)=\mu(I/I^2)+1$.

Question

Let $I$ be a finitely generated ideal of a commutative ring $R$ (with unity 1). Let $\mu(S)$ denotes the cardinality of minimal generating set of $S$. It can be shown (via determinant trick) that $$\mu(I/I^2)\le \mu(I) \le \mu(I/I^2)+1$$

What I wish to see is few examples where $\mu(I)=\mu(I/I^2)+1$.

I was thinking about a simple case where $I$ is a non principal ideal of Noetherian ring but $I/I^2$ is cyclic as an $R/I$ module. But didn't find any. Any help / suggestions.

Answer 1

Just take $I$ to be a non-principal prime ideal in a Dedekind domain.

As a concrete example, how about $R=\Bbb Z[\sqrt{-6}]$ and $I=\left<2,\sqrt{-6}\right>$?

Answer 2

If $I$ is generated by a (nonzero) idempotent, then $I^2=I$, so $\mu(I)=1$ but $\mu(I/I^2)=0$. For an explicit example, you could take $R=\mathbb{Z}\times\mathbb{Z}$ and $I$ to be the ideal generated by $(1,0)$.