Example of an invertible ideal which contains a zero divisor as one of its generators

Question

Let $R$ be a ring and let $I$ be an (integral) invertible ideal of $R$. Let $J = \{a_1,\ldots,a_r\}$ be a minimal ideal generating set of $I$ in the sense that any proper subset of $J$ will not generate $I$.

I was wondering if it was possible that one of the $a_i$ was a zero divisor of $R$.

Clearly, any invertible ideal does indeed contain at least one regular element and hence any principal invertible ideal won't provide such an example.

Thanks a lot in advance!

Answer 1

Yes, it is quite possible.
Let $k$ be a field, $R=k\times k$ and $I=R$.
Then $a_1=(1,0), a_2=(0,1)$ form a minimal set of generators for $I=R=\langle a_1, a_2\rangle$ but both $a_1,a_2$ are zero divisors of $R$.

Nota Bene
Contrary to what you claim this minimal set of generators contains no regular element.