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.

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.
• If the original ring is a local ring and the ideal in question is the unique maximal ideal, then the natural homomorphism from the ring to its completion is injective. This is a consequence of the Krull intersection theorem for Jacobson radical. Further information: Natural homomorphism to completion is injective for Noetherian local ring
• If the original ring is an integral domain, it injects into its completion. This follows from the Krull intersection theorem for Noetherian domains.
• The completion of a ring at a maximal ideal is naturally isomorphic to the completion of its localization at the maximal ideal, at the unique maximal ideal of that local ring.