A pseudocompact ring is a complete Hausdorff topological ring, , which admits a base at of two-sided open ideals for each of which is an Artinian ring. Equivalently, it is a regular pro-object in Artinian rings.
More generally let be a commutative pseudocompact ring. A complete Hausdorff topological ring will be called a pseudocompact algebra over if
(i) is an algebra in the usual sense, and
(ii) admits a system of open neighbourhoods of 0 consisting of two-sided ideals such that has finite length as an module.
Grothendieck developed the theory of formal groups over pseudocompact rings.
The homological algebra of such rings and the corresponding modules are discussed in