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2. Integral domain - Wikipedia

   en.wikipedia.org/wiki/Integral_domain

   An inductive limit of integral domains is an integral domain. If A , B are integral domains over an algebraically closed field k , then A ⊗ k B is an integral domain. This is a consequence of Hilbert's nullstellensatz , [ a ] and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic ...

3. Domain (ring theory) - Wikipedia

   en.wikipedia.org/wiki/Domain_(ring_theory)

   In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.

4. Valuation ring - Wikipedia

   en.wikipedia.org/wiki/Valuation_ring

   In abstract algebra, a valuation ring is an integral domain D such that for every non-zero 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 every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F.

5. Principal ideal domain - Wikipedia

   en.wikipedia.org/wiki/Principal_ideal_domain

   An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals. An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.

6. Field of fractions - Wikipedia

   en.wikipedia.org/wiki/Field_of_fractions

   In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.

7. Ring (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Ring_(mathematics)

   A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.)

8. Discrete valuation ring - Wikipedia

   en.wikipedia.org/wiki/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 that satisfies any and all of the following equivalent conditions: R is a local ring, a principal ideal domain, and not a field.

9. Unique factorization domain - Wikipedia

   en.wikipedia.org/wiki/Unique_factorization_domain

   A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain. In general, for an integral domain A, the following conditions are equivalent: A is a UFD.