Wikipedia / Overring
Wikipedia / Overring
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.
is a subring;.
Properties
Ring of fractions
are the rings of fractions by multiplicative set. is the total ring of fractions if every nonunit is a zero-divisor.. is integrally closed.
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.
is locally nilpotentfree 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 (a finitely generated algebra) 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.
.
is a Noetherian ring.
is a Noetherian ring.
is locally nilpotentfree with restricted dimension 1 or less.
has restricted dimension 1 or less.
is integrally closed.
.
is locally nilpotentfree.
-module.
is Noetherian.
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 an integral domain extension.
is an intermediate.
Properties
A Noetherian ring's Krull dimension is 1 or less if every overring is coherent.
.
is a Prüfer domain if each proper 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.
is a Prüfer domain is equivalent to:
is the intersection is integrally closed.
is integrally closed.
is integrally closed.
is integrally closed
is integrally closed.
is coherent.
is a Prüfer domain is equivalent to:
is flat-module.
Each valuation overring is a ring of fractions.
Minimal overring
Definitions
A minimal ring homomorphism is an injective non-surjective is an isomorphism.
A proper minimal ring extension occurs if the ring inclusion has no proper intermediate ring.
A minimal overring has no proper intermediate ring.
The Kaplansky ideal transform (Hayes transform, S-transform.
Properties
is not a field.
is not a field.
.
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
Notes
References
Further reading