User:TMM53/overrings-2023-03-16

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 $B$ is an overring of ring $A$  if $A$  is a subring of $B$  and $B$  is a subring of the total ring of fractions $Q(A)$ ; the relationship is $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.

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

Theorems
Assume $T$ is an overring of $R$  and $A$  is a multiplicative system and $A \subset R \ \backslash \ \{0 \}$. The implications are:
 * The ring $T_{A}$ is an overring of $R_{A}$ .  The ring $T_{A}$  is the total ring of fractions of $R_{A}$  if every  nonunit element of $T_{A}$  is a zero-divisor.
 * Every overring of $R_{A}$ contained in $T_{A}$  is a ring $S_{A}$, and $S$  is an overring of $R$.
 * Ring $R_{A}$ is integrally closed in $T_{A}$  if $R$  is integrally closed in $T$.

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

These statements are equivalent for affine ring $R$ with integral closure $\bar{R}$.
 * Ring $R$ is locally nilpotentfree.
 * Ring $\bar{R}$ is a finite $\operatorname{R -}$ module.
 * Ring $\bar{R}$ is Noetherian.

An integrally closed local ring $R$ 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.

Definitions
A coherent ring is a commutative ring with each finitely generated ideal finitely presented. Noetherian domains and Prüfer domains are coherent.

A pair $(R,T)$ indicates that $T$  is an integral domain extension over $R$  with $R \subseteq T$.

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

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

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

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

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

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 $R$ is a Prüfer domain is equivalent to:
 * Each overring of $ R$  is the intersection of localizations of $ R$,   and $ R$  is integrally closed.
 * Each overring of $ R$  is the intersection of rings of fractions of $ R$,   and $ R$  is integrally closed.
 * Each overring of $ R$  has prime ideals that are extensions of the prime ideals of $ R$, and $ R$  is integrally closed.
 * Each overring of $ R$  has at most 1 prime ideal lying over any prime ideal of $ R$,   and $ R$  is integrally closed
 * Each overring of $ R$  is integrally closed.
 * Each overring of $ R$  is coherent.

The statement $R$ is a Prüfer domain is equivalent to:
 * Each overring S of $R$ is flat as a $$\operatorname{S-}$$module.
 * Each valuation overring of $R$ is a ring of fractions.

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

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

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

The Kaplansky ideal transform ( Hayes transform, S-transform ) for ideal $I$ in ring $R$  is:
 * $S_{R}(I)= \{ x \in Q(R) \ | \ \forall y \in I, \ \exists n \in \mathbb{N} \wedge x\cdot y^{n} \in R \}$

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

The field of fractions of $R$ contains minimal overring $T$  of $R$  when $R$  is not a field.

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

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.

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