Completion (ring theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have simpler structure than the general ones and Hensel's lemma applies to them. Geometrically, a completion of a commutative ring R concentrates on a formal neighborhood of a point or a Zariski closed subvariety of its spectrum Spec R.
Suppose that E is an abelian group with a descending filtration
of subgroups. One then defines the completion (with respect to the filtration) as the inverse limit:
This is again an abelian group. Usually E is an additive abelian group. If E has additional algebraic structure compatible with the filtration, for instance E is a filtered ring, a filtered module, or a filtered vector space, then its completion is again an object with the same structure that is complete in the topology determined by the filtration. This construction may be applied both to commutative and noncommutative rings. As may be expected, when the intersection of the equals zero, this produces a complete topological ring.
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