separability of commutative rings

Before discussing on the main Question I should recall two notions in the area of commutative rings.

By $Max(R)$, we mean the set of all maximal ideals of the commutative unitary ring $R$.

Definition 1: A commutative ring $R$ is called separable, if for any subset $(M_{\alpha})_{\alpha \in A} \subset Max(R)$ with $\cap _{\alpha \in A}M _{\alpha}=(0)$, there exists a countable subset $A_0 \subset A$ that $\cap _{\alpha \in A _{0}}M _{\alpha}=(0)$.

and

Definition 2: A commutative ring $R$ is called $\aleph_{1}$-cogenerated if for any collection $(I_{\alpha})_{\alpha \in A}$, with $\cap _{\alpha \in A} I _{\alpha}=(0)$ there exists a countable subset $A_0 \subset A$ so that $\cap _{\alpha \in A _{0}}I _{\alpha}=(0)$.

It is clear that every $\aleph_1$-cogenerated ring is separable. I am looking for a commutative $J$-semisimple ring $R$ $($i.e. $J(R)=0)$ that is a separable ring but is not $\aleph_1$-cogenerated.

Question: Is there a commutative $J$-semisimple ring $R$ that is separable but is not $\aleph_1$-cogenerated.

Addendum: The ring $C(X)$ of all continuous real valued functions on a topological space $X$ is an example of a commutative $J$-semisimple ring for which separability and $\aleph_1$-cogenerated are equivalent.

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