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Local ring

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A ring A is said to be a local ring if it has a unique maximal ideal m. It is said to be semi-local if it has finitely many maximal ideals.

The localisation of a commutative integral domain at a non-zero prime ideal is a local ring.

Properties

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 \bigcap_n m^n = \{0\} and A is complete with respect to the uniformity defined by the cosets of the powers of m.

References

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