If $R\in \mathbf{CRing}$, and $(I_n)$ is a finite family of ideals, we can define the product of this family to be the set of finite sums of products $x_1 \cdots x_n$ where each $x_j \in I_j$. This notion obviously makes no sense for infinite families since we don't have a notion of infinite product. So it seems natural to try generalizing it for topological (commutative) rings. Now if $R\in \mathbf{CRing(Top)}$ and $(I_n)_{n\in \mathbb N}$, we could define the product of that family as the set of finite sums of limits of form $\prod_{n\in \mathbb N} x_n$ where each $x_n \in I_n$ (ignoring non-convergent ones obviously). It is easy to see this is an ideal.
Has this notion been used? What are some references that use this?
(It can also probably be more generalized using directed sets but the writter and probably many readers are not as experienced with those as with limits of sequences.)