Completion of a ring

Definition

Symbol-free definition

Let $R$ be a commutative unital ring and $I$ be a maximal ideal inside $R$. The completion of $R$ with respect to the ideal $I$ is defined as the inverse limit of the factor rings $R/I^{j}$ under the natural quotient maps.

Note that there is a natural embedding of $R$ inside its completion.

A ring is said to be complete with respect to a maximal ideal if the map to its completion with respect to that ideal is an isomorphism.

Facts

The completion with respect to a maximal ideal turns out to be a local ring whose residue field is the same as the quotient by this maximal ideal. In fact, the maximal ideal for the completion is the ideal generated by the maximal ideal in the original ring, over the completion.