Ideals of $\prod_j R$ is of the form $\prod_j I_j$?

Question

Let $R$ be a commutative ring with unity. Let $S = \prod_{i \in \mathbb{N}} R$. Is every ideal of $S$ of the form $\prod_{i \in \mathbb{N}} I_i$, where $I_i$ is an ideal of $R$? I think it's true.... but I wasn't too sure. Any comments or explanations are appreciated. Thank you.

Answer 1

The conclusion holds for finitely many factors, but does not hold for infinitely many factors (whether it is the same ring or a family of rings different from each other).

Consider the ideal of all tuples $(x_i)$ such that $x_j=0$ for all but finitely many $j$ (that is, $\oplus_{i\in\mathbb{N}} R_i$). This is easily verified to be an ideal, but it cannot be of the form you give, since the projection on each component is equal to all of $R_j$, and the only ideal of the form $\prod I_j$ with all projections equal to the whole ring is $\prod R_j$ itself.