User:TMM53/overrings-2023-03-16

This page is about the mathematical concept. For the accent, see ring (diacritic).

Overrings are common in algebra. Intuitively, an overring contains a ring. For example, the overring-to-ring relationship is similar to the fraction to integer relationship. Among all integer fractions, the fractions with a 1 denominator correspond to the integers. Overrings are important because they help us better understand the properties of different types of rings and domains.

Definition

Ring ${\textstyle B}$ is an overring of ring ${\textstyle A}$ if ${\textstyle A}$ is a subring of ${\textstyle B}$ and ${\textstyle B}$ is a subring of the total ring of fractions ${\textstyle Q(A)}$; the relationship is ${\textstyle A\subseteq B\subseteq Q(A)}$.

Properties

Unless otherwise stated, all rings are commutative rings, and each ring and its overring share the same identity element.

Ring of fractions

Definitions

The ring ${\textstyle R_{A}}$ is the ring of fractions (ring of quotients, localization) of ring ${\textstyle R}$ by multiplicative system set ${\textstyle A}$, ${\textstyle A\subset R\ \backslash \ \{0\}}$.

Theorems

Assume ${\textstyle T}$ is an overring of ${\textstyle R}$ and ${\textstyle A}$ is a multiplicative system and ${\textstyle A\subset R\ \backslash \ \{0\}}$. The implications are:

• The ring ${\textstyle T_{A}}$ is an overring of ${\textstyle R_{A}}$. The ring ${\textstyle T_{A}}$ is the total ring of fractions of ${\textstyle R_{A}}$ if every nonunit element of ${\textstyle T_{A}}$ is a zero-divisor.
• Every overring of ${\textstyle R_{A}}$ contained in ${\textstyle T_{A}}$ is a ring ${\textstyle S_{A}}$, and ${\textstyle S}$ is an overring of ${\textstyle R}$.
• Ring ${\textstyle R_{A}}$ is integrally closed in ${\textstyle T_{A}}$ if ${\textstyle R}$ is integrally closed in ${\textstyle T}$.

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 ${\textstyle R}$ is locally nilpotentfree if every ${\textstyle R_{M}}$, generated by each maximal ideal ${\textstyle M}$, is free of nilpotent elements or a ring with every non-unit a zero divisor.

An affine ring is the homomorphic image of a polynomial ring over a field.

The torsion class group of a Dedekind domain is the group of fractional domains modulo the principal fractional ideals subgroup.

Theorems

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 ${\textstyle R}$ with integral closure ${\textstyle {\bar {R}}}$.

• Every overring of ${\textstyle R}$ is a Noetherian ring.
• For each maximal ideal ${\textstyle M}$ of ${\textstyle R}$, every overring of ${\textstyle R_{M}}$ is a Noetherian ring.
• Ring ${\textstyle R}$ is locally nilpotentfree with restricted dimension 1 or less.
• Ring ${\textstyle {\bar {R}}}$ is Noetherian, and ring ${\textstyle R}$ has restricted dimension 1 or less.
• Every overring of ${\textstyle {\bar {R}}}$ is integrally closed.

These statements are equivalent for affine ring ${\textstyle R}$ with integral closure ${\textstyle {\bar {R}}}$.

• Ring ${\textstyle R}$ is locally nilpotentfree.
• Ring ${\textstyle {\bar {R}}}$ is a finite ${\textstyle \operatorname {R-} }$module.
• Ring ${\textstyle {\bar {R}}}$ is Noetherian.

An integrally closed local ring ${\textstyle R}$ is an integral domain or a ring whose non-unit elements are all zero-divisors.

A Noetherian integral domain is a Dedekind ring if and only if every overring of the Noetherian ring is integrally closed.

Every overring of a Noetherian integral domain is a ring of fractions if and only 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 ${\textstyle (R,T)}$ indicates that ${\textstyle T}$ is an integral domain extension over ${\textstyle R}$ with ${\textstyle R\subseteq T}$.

An intermediate domain ${\textstyle S}$ for pair ${\textstyle (R,T)}$ indicates this relationship ${\textstyle R\subseteq S\subseteq T}$.

Theorems

A Noetherian ring's Krull dimension is 1 or less if every overring is coherent.

For integral domain pair ${\textstyle (R,T)}$, ${\textstyle T}$ is an overring of ${\textstyle R}$ if each intermediate integral domain is integrally closed in ${\textstyle T}$.

The integral closure of ${\textstyle R}$ is a Prüfer domain if each proper overring of ${\textstyle R}$ is coherent.

The overrings of Prüfer domains and Krull 1-dimensional Noetherian domains are coherent.

Prüfer domains

Theorems

A ring has QR property if every overring is a localization with a multiplicative system.

• 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 and only if the radical of every finitely generated ideal equals the radical generated by a principal ideal.

The statement ${\textstyle R}$ is a Prüfer domain is equivalent to:

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

The statement ${\textstyle R}$ is a Prüfer domain is equivalent to:

• Each overring $S$ of ${\textstyle R}$ is flat as a $\operatorname {S-}$module.
• Each valuation overring of ${\textstyle R}$ is a ring of fractions.

Minimal overring

Definitions

A minimal ring homomorphism ${\textstyle f:R\hookrightarrow T}$ is an injective non-surjective homomorophism, and any decomposition ${\textstyle f=g\circ h}$ implies ${\textstyle g}$ or ${\textstyle h}$ is an isomorphism.

A proper minimal ring extension ${\textstyle T}$ of subring ${\textstyle R}$ occurs when the ring inclusion ${\textstyle f:R\hookrightarrow T}$ is a minimal ring homomorphism. This implies the ring pair ${\textstyle (R,T)}$ has no proper intermediate ring.

A minimal overring integral domain ${\textstyle T}$ of integral domain ${\textstyle R}$ occurs when ${\textstyle T}$ contains ${\textstyle R}$ as a subring, and the ring pair ${\textstyle (R,T)}$ has no proper intermediate ring.

The Kaplansky ideal transform (Hayes transform, S-transform) for ideal ${\textstyle I}$ in ring ${\textstyle R}$ is:

${\textstyle S_{R}(I)=\{x\in Q(R)\ |\ \forall y\in I,\ \exists n\in \mathbb {N} \wedge x\cdot y^{n}\in R\}}$

Theorems

Any domain generated from a minimal ring extension of domain ${\textstyle R}$ is an overring of ${\textstyle R}$ if ${\textstyle R}$ is not a field. The 1st of 3 types of minimal ring extensions ${\textstyle S}$ of domain ${\textstyle R}$ generates a domain and minimal overring ${\textstyle S}$ of ${\textstyle R}$ that contains ${\textstyle R}$.

The field of fractions of ${\textstyle R}$ contains minimal overring ${\textstyle T}$ of ${\textstyle R}$ when ${\textstyle R}$ is not a field.

If a minimal overring of a non-field integrally closed integral domain ${\textstyle R}$ exists, this minimal overring occurs as the Kaplansky transform of a maximal ideal of ${\textstyle R}$.

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 denominator. The dyadic rational ring is the localization of the integers by powers of two and an overring of the integer ring.

References

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• Bazzoni, Silvana; Glaz, Sarah (2006). "Prüfer rings". In Brewer rings, James W.; Glaz, Sarah; Heinzer, William J.; Olberding, Bruce M. (eds.). Multiplicative ideal theory in commutative algebra: a tribute to the work of Robert Gilmer. New York, NY: Springer. pp. 54–72. ISBN 978-0-387-24600-0.
• Cohen, Irving S. (1950). "Commutative rings with restricted minimum condition". Duke Math. J. 17 (1): 27–42. doi:10.1215/S0012-7094-50-01704-2.
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Related categories

Category:Ring theory Category:Algebraic structures Category:Commutative algebra