Local ring

A local ring is a ring with exactly one maximal ideal. This condition gives its elements a simple unit criterion: every element outside the maximal ideal is invertible.

Ring (mathematics): A ring is a set with addition and multiplication satisfying familiar algebraic laws, including distributivity. Local rings are rings distinguished by the number of their maximal ideals.
Residue field: A residue field is the quotient of a local ring by its maximal ideal. It captures the field-valued information left after the maximal ideal is collapsed.
Localization at a prime ideal: Localization at a prime ideal inverts every ring element outside that ideal. It creates the local ring used to study algebraic behavior near that prime.
Semilocal ring: A semilocal ring is a ring with finitely many maximal ideals. It relaxes the defining condition from one maximal ideal to finitely many.
Ideal (ring theory): An ideal is an additive subgroup of a ring closed under multiplication by ring elements. The unique maximal ideal is a particular ideal with no larger proper ideal.
Jacobson radical: The Jacobson radical of a ring is the intersection of all its maximal left ideals. For a commutative local ring, this radical is exactly its unique maximal ideal.
Local ring of an algebraic variety: The local ring of an algebraic variety at a point consists of functions defined near that point, identified when they agree nearby. Its unique maximal ideal records functions vanishing at the chosen point.
Artinian ring: An Artinian ring satisfies the descending chain condition on ideals. Artinian rings need not be local, though every commutative Artinian ring decomposes into local pieces.
Maximal ideal: A maximal ideal is a proper ideal not contained in any larger proper ideal. A local ring is defined by having exactly one maximal ideal.
Nakayama's lemma: Nakayama's lemma states conditions under which a finitely generated module over a ring is zero, using its Jacobson radical. It turns containment in the maximal ideal into a powerful test for finite modules.

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