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From Citizendium, the Citizens' Compendium
A ring is said to be a local ring if it has a unique maximal ideal . It is said to be semi-local if it has finitely many maximal ideals.
In a local ring the unit group is the complement of the maximal ideal.
Complete local ring
A local ring A is complete if the intersection and A is complete with respect to the uniformity defined by the cosets of the powers of m.