Non-local ring $A$ with an ideal $I$ such that $1 + I \subseteq A^{*}$

Question

It's well-known that $A$ is local iff there exists a maximal ideal $\mathfrak{m} \subseteq A$ such that $1 + \mathfrak{m} \subseteq A^{*}$.

I'm looking for an ideal $I$ not maximal such that $1 + I$ is invertible and $A$ being non-local. Thanks!

Answer 1

On a lighter side, why not take $I=0$ in any ring?

For a slightly better one, take $A=k[x,y]/(x^2)$ and $I=xA$, $k$ any field, say.