A pseudocompact ring is a complete Hausdorff topological ring, $R$, which admits a base at $0$ of two-sided open ideals $I$ for each of which $R/I$ is an Artinian ring. Equivalently, it is a regular pro-object in Artinian rings.

More generally let $R$ be a commutative pseudocompact ring. A complete Hausdorff topological ring $\Lambda$ will be called a pseudocompact algebra over $R$ if

(i) $\Lambda$ is an $R$ algebra in the usual sense, and

(ii) $\Lambda$ admits a system of open neighbourhoods of 0 consisting of two-sided ideals $I$ such that $\Lambda/I$ has finite length as an $R$ module.

Motivations and applications

Grothendieck developed the theory of formal groups over pseudocompact rings.