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pseudocompact ring

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

1. Grothendieck developed the theory of formal groups over pseudocompact rings.

2. The completed group algebras of profinite groups are naturally pseudocompact algebras, usually considered over finite fields or the profinite completion of the ring of integers.

References

The homological algebra of such rings and the corresponding modules are discussed in

• A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966) 442-470, MR202790, doi