Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring.
Overrings provide an improved understanding of different types of rings and domains.
represent the field of fractions of an integral domain
[4]: 52–53 A Noetherian ring satisfies the 3 equivalent finitenss conditions i) every ascending chain of ideals is finite, ii) every non-empty family of ideals has a maximal element and iii) every ideal has a finite basis.
[3]: 270 A ring's restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element.
is free of nilpotent elements or a ring with every nonunit a zero divisor.
[4]: 52 An affine ring is the homomorphic image of a polynomial ring over a field.
[5][6] Every overrring of a direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.
[4]: 53 These statements are equivalent for Noetherian ring
[4]: 57 These statements are equivalent for affine ring
[4]: 58 An integrally closed local ring
is an integral domain or a ring whose non-unit elements are all zero-divisors.
[4]: 58 A Noetherian integral domain is a Dedekind ring if every overring of the Noetherian ring is integrally closed.
[7]: 198 Every overring of a Noetherian integral domain is a ring of fractions if the Noetherian integral domain is a Dedekind ring with a torsion class group.
[7]: 200 A coherent ring is a commutative ring with each finitely generated ideal finitely presented.
[9]: 331 A Noetherian ring's Krull dimension is 1 or less if every overring is coherent.
[2]: 373 For integral domain pair
is a Prüfer domain if each proper overring of
[8]: 138 A ring has QR property if every overring is a localization with a multiplicative set.
[11]: 196 A Prüfer domain is a QR domain if the radical of every finitely generated ideal equals the radical generated by a principal ideal.
is a Prüfer domain is equivalent to:[13]: 56 The statement
is a Prüfer domain is equivalent to:[1]: 167 A minimal ring homomorphism
is an injective non-surjective homomorophism, and if the homomorphism
[14]: 461 A proper minimal ring extension
[16]: 60 The Kaplansky ideal transform (Hayes transform, S-transform) of ideal
is not a field, If a minimal overring of integral domain
exists, this minimal overring occurs as the Kaplansky transform of a maximal ideal of
[16]: 60 The Bézout integral domain is a type of Prüfer domain; the Bézout domain's defining property is every finitely generated ideal is a principal ideal.
[7]: 196 The dyadic rational is a fraction with an integer numerator and power of 2 denominators.
The dyadic rational ring is the localization of the integers by powers of two and an overring of the integer ring.