I am looking for a commutative ring $R$ with identity that has the following properties:

1) $\mathrm{Max}(R)$, the set of all maximal ideals of $R$, is infinite;

2) whenever $\{I_\alpha\}_{\alpha\in\Gamma}$ is an arbitrary family of finitely generated ideals of $R$ such that for each maximal ideal $\mathfrak m$ there exists $\beta\in\Gamma$ with $I_\beta\subseteq\mathfrak m$, then there exists a finite subfamily $\{I_1, I_2,..., I_n\}$ of $\{I_\alpha\}_{\alpha\in\Gamma}$ with $I_1 I_2\cdots I_n\subseteq J(R)$.

Here $J(R)$ is the Jacobson radical, the intersection of all maximal ideals of $R$.