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.

Definition

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In this article, all rings are commutative rings, and ring and overring share the same identity element.

Let   represent the field of fractions of an integral domain  . Ring   is an overring of integral domain   if   is a subring of   and   is a subring of the field of fractions  ;[1]: 167  the relationship is  .[2]: 373 

Properties

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Ring of fractions

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The rings   are the rings of fractions of rings   by multiplicative set  .[3]: 46  Assume   is an overring of   and   is a multiplicative set in  . The ring   is an overring of  . The ring   is the total ring of fractions of   if every nonunit element of   is a zero-divisor.[4]: 52–53  Every overring of   contained in   is a ring  , and   is an overring of  .[4]: 52–53  Ring   is integrally closed in   if   is integrally closed in  .[4]: 52–53 

Noetherian domain

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Definitions

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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]: 199 

An integral domain is a Dedekind domain if every ideal of the domain is a finite product of prime ideals.[3]: 270 

A ring's restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element.[4]: 52 

A ring   is locally nilpotentfree if every ring   with maximal ideal   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.[4]: 58 

Properties

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Every overring of a Dedekind ring is a Dedekind ring.[5][6]

Every overrring of a direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.[4]: 53 

Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.[4]: 53 

These statements are equivalent for Noetherian ring   with integral closure  .[4]: 57 

  • Every overring of   is a Noetherian ring.
  • For each maximal ideal   of  , every overring of   is a Noetherian ring.
  • Ring   is locally nilpotentfree with restricted dimension 1 or less.
  • Ring   is Noetherian, and ring   has restricted dimension 1 or less.
  • Every overring of   is integrally closed.

These statements are equivalent for affine ring   with integral closure  .[4]: 58 

  • Ring   is locally nilpotentfree.
  • Ring   is a finite  module.
  • Ring   is Noetherian.

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 

Coherent rings

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Definitions

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A coherent ring is a commutative ring with each finitely generated ideal finitely presented.[2]: 373  Noetherian domains and Prüfer domains are coherent.[8]: 137 

A pair   indicates a integral domain extension of   over  .[9]: 331 

Ring   is an intermediate domain for pair   if   is a subdomain of   and   is a subdomain of  .[9]: 331 

Properties

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A Noetherian ring's Krull dimension is 1 or less if every overring is coherent.[2]: 373 

For integral domain pair  ,   is an overring of   if each intermediate integral domain is integrally closed in  .[9]: 332 [10]: 175 

The integral closure of   is a Prüfer domain if each proper overring of   is coherent.[8]: 137 

The overrings of Prüfer domains and Krull 1-dimensional Noetherian domains are coherent.[8]: 138 

Prüfer domains

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Properties

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A ring has QR property if every overring is a localization with a multiplicative set.[11]: 196  The QR domains are Prüfer domains.[11]: 196  A Prüfer domain with a torsion Picard group is a QR domain.[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.[12]: 500 

The statement   is a Prüfer domain is equivalent to:[13]: 56 

  • Each overring of   is the intersection of localizations of  , and   is integrally closed.
  • Each overring of   is the intersection of rings of fractions of  , and   is integrally closed.
  • Each overring of   has prime ideals that are extensions of the prime ideals of  , and   is integrally closed.
  • Each overring of   has at most 1 prime ideal lying over any prime ideal of  , and   is integrally closed
  • Each overring of   is integrally closed.
  • Each overring of   is coherent.

The statement   is a Prüfer domain is equivalent to:[1]: 167 

  • Each overring   of   is flat as a  module.
  • Each valuation overring of   is a ring of fractions.

Minimal overring

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Definitions

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A minimal ring homomorphism   is an injective non-surjective homomorophism, and if the homomorphism   is a composition of homomorphisms   and   then   or   is an isomorphism.[14]: 461 

A proper minimal ring extension   of subring   occurs if the ring inclusion of   in to   is a minimal ring homomorphism. This implies the ring pair   has no proper intermediate ring.[15]: 186 

A minimal overring   of ring   occurs if   contains   as a subring, and the ring pair   has no proper intermediate ring.[16]: 60 

The Kaplansky ideal transform (Hayes transform, S-transform) of ideal   with respect to integral domain   is a subset of the fraction field  . This subset contains elements   such that for each element   of the ideal   there is a positive integer   with the product   contained in integral domain  .[17][16]: 60 

Properties

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Any domain generated from a minimal ring extension of domain   is an overring of   if   is not a field.[17][15]: 186 

The field of fractions of   contains minimal overring   of   when   is not a field.[16]: 60 

Assume an integrally closed integral domain   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 

Examples

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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. The Bézout domain will share all the overring properties of a Prüfer domain.[1]: 168 

The integer ring is a Prüfer ring, and all overrings are rings of quotients.[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.

See also

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Notes

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References

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