Overring
Introduction 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 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 ; the relationship is .
Properties ===Ring of fractions===The rings  are the rings of fractions of rings  by multiplicative set . 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. Every overring of  contained in  is a ring , and  is an overring of . Ring  is integrally closed in  if  is integrally closed in .===Noetherian domain=======Definitions====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. An integral domain is a Dedekind domain if every ideal of the domain is a finite product of prime ideals.A ring's restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element. 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. An affine ring is the homomorphic image of a polynomial ring over a field. ====Properties====Every overring of a Dedekind ring is a Dedekind ring.Every overrring of a direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.These statements are equivalent for Noetherian ring  with integral closure . * 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 . * 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. A Noetherian integral domain is a Dedekind ring if every overring of the Noetherian ring is integrally closed. 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.===Coherent rings=======Definitions====A coherent ring is a commutative ring with each finitely generated ideal finitely presented. Noetherian domains and Prüfer domains are coherent. A pair  indicates a integral domain extension of  over . Ring  is an intermediate domain for pair  if  is a subdomain of  and  is a subdomain of . ====Properties====A Noetherian ring's Krull dimension is 1 or less if every overring is coherent.For integral domain pair ,  is an overring of  if each intermediate integral domain is integrally closed in . The integral closure of  is a Prüfer domain if each proper overring of  is coherent. The overrings of Prüfer domains and Krull 1-dimensional Noetherian domains are coherent. ===Prüfer domains=======Properties====A ring has QR property if every overring is a localization with a multiplicative set. The QR domains are Prüfer domains. A Prüfer domain with a torsion Picard group is a QR domain. A Prüfer domain is a QR domain if the radical of every finitely generated ideal equals the radical generated by a principal ideal.The statement  is a Prüfer domain is equivalent to:* 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:* Each overring  of  is flat as a module. * Each valuation overring of  is a ring of fractions.===Minimal overring=======Definitions====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.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. A minimal overring  of ring  occurs if  contains  as a subring, and the ring pair  has no proper intermediate ring. 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 .====Properties====Any domain generated from a minimal ring extension of domain  is an overring of  if  is not a field. The field of fractions of  contains minimal overring  of  when  is not a field. 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 .
Examples 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. The integer ring is a Prüfer ring, and all overrings are rings of quotients. 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 * * * * * Glossary of ring theory* * * * Localization (commutative algebra)* * * * *  * Regular element (in ring theory):** ** * * *
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