Let $A$ be a ring. Suppose we have a partially ordered set $X$ of nonzero ideals of $A$, which is closed under intersection, summation and product of ideals. We can consider $\text{lim}_{I \in X} A /I$ (inverse limit). This generalizes the situation where $X = \{ A, I, I^2, I^3, ... \}$ for a nonzero ideal $I$.
Do people still consider this inverse limit? I am wondering about its structure. For instance, if $X$ is totally ordered, then do we have a theorem like Cohen's structure theorem but replacing "discrete valuation ring" with "valuation ring".
Sure. Here's a nice example: if $A$ is an algebra over a field $k$, we can consider the cofiltered limit over all finite-dimensional quotients $A/I$ (so $I$ ranges over all "cofinite" two-sided ideals). This is an algebra that deserves to be called the profinite completion $\widehat{A}$ of $A$. It appears naturally, for example, as the endomorphism algebra of the forgetful functor $\text{Mod}_f(A) \to \text{Vect}_f$ from finite-dimensional $A$-modules to finite-dimensional vector spaces. It also appears when considering the relationship between categories of modules over algebras and categories of comodules over coalgebras; see, for example, this math.SE question.