discrete valuation ring

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• Discrete valuation ring — In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non zero maximal ideal. This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions: R is a local… …   Wikipedia

• Valuation ring — In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F , at least one of x or x 1 belongs to D .Given a field F , if D is a subring of F such that either x or x 1 belongs to D for… …   Wikipedia

• Discrete valuation — In mathematics, a discrete valuation is an integer valuation on a field k, that is a function satisfying the conditions . Note that often the trivial valuation which takes on only the values …   Wikipedia

• Ring (mathematics) — This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… …   Wikipedia

• Ring of mixed characteristic — In commutative algebra, a ring of mixed characteristic is a commutative ring R having characteristic zero and having an ideal I such that R / I has positive characteristic. Examples The integers Z have characteristic zero, but for any prime… …   Wikipedia

• Valuation (algebra) — In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. They generalize to commutative algebra the notion of size… …   Wikipedia

• Local ring — In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called local behaviour , in the sense of functions defined on varieties or manifolds, or of… …   Wikipedia

• Regular local ring — In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. The minimal number of generators of the maximal… …   Wikipedia

• Krull ring — A Krull ring is a particular type of commutative ring studied in commutative algebra and related branches of mathematics and named after the German mathematician Wolfgang Krull.Formal definitionLet A be an integral domain and let P be the set of… …   Wikipedia

• Adele ring — In number theory, the adele ring is a topological ring which is built on the field of rational numbers (or, more generally, any algebraic number field). It involves all the completions of the field.The word adele is short for additive idele .… …   Wikipedia

• Local field — In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non discrete topology.[1] Given such a field, an absolute value can be defined on it. There are two basic types of local field …   Wikipedia