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

Let ${\textstyle Q(A)}$  represent the field of fractions of an integral domain ${\textstyle A}$ . Ring ${\textstyle B}$  is an overring of integral domain ${\textstyle A}$  if ${\textstyle A}$  is a subring of ${\textstyle B}$  and ${\textstyle B}$  is a subring of the field of fractions ${\textstyle Q(A)}$ ;^[1]: 167 the relationship is ${\textstyle A\subseteq B\subseteq Q(A)}$ .^[2]: 373

Ring of fractions edit

The rings ${\textstyle R_{A},S_{A},T_{A}}$  are the rings of fractions of rings ${\textstyle R,S,T}$  by multiplicative set ${\textstyle A}$ .^[3]: 46 Assume ${\textstyle T}$  is an overring of ${\textstyle R}$  and ${\textstyle A}$  is a multiplicative set in ${\textstyle R}$ . 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.^{[4]: 52–53} Every overring of ${\textstyle R_{A}}$  contained in ${\textstyle T_{A}}$  is a ring ${\textstyle S_{A}}$ , and ${\textstyle S}$  is an overring of ${\textstyle R}$ .^{[4]: 52–53} Ring ${\textstyle R_{A}}$  is integrally closed in ${\textstyle T_{A}}$  if ${\textstyle R}$  is integrally closed in ${\textstyle T}$ .^{[4]: 52–53}

Noetherian domain edit

Definitions edit

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 ${\textstyle R}$  is locally nilpotentfree if every ring ${\textstyle R_{M}}$  with maximal ideal ${\textstyle M}$  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 edit

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 ${\textstyle R}$  with integral closure ${\textstyle {\bar {R}}}$ .^[4]: 57

• 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 ${\textstyle {\bar {R}}}$ .^[4]: 58

• 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.^[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 edit

Definitions edit

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 ${\textstyle (R,T)}$  indicates a integral domain extension of ${\textstyle T}$  over ${\textstyle R}$ .^[9]: 331

Ring ${\textstyle S}$  is an intermediate domain for pair ${\textstyle (R,T)}$  if ${\textstyle R}$  is a subdomain of ${\textstyle S}$  and ${\textstyle S}$  is a subdomain of ${\textstyle T}$ .^[9]: 331

Properties edit

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

For integral domain pair ${\textstyle (R,T)}$ , ${\textstyle T}$  is an overring of ${\textstyle R}$  if each intermediate integral domain is integrally closed in ${\textstyle T}$ .^{[9]: 332 [10]: 175}

The integral closure of ${\textstyle R}$  is a Prüfer domain if each proper overring of ${\textstyle R}$  is coherent.^[8]: 137

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

Prüfer domains edit

Properties edit

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 ${\textstyle R}$  is a Prüfer domain is equivalent to:^[13]: 56

• 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:^[1]: 167

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

Minimal overring edit

Definitions edit

A minimal ring homomorphism ${\textstyle f}$  is an injective non-surjective homomorophism, and if the homomorphism ${\textstyle f}$  is a composition of homomorphisms ${\textstyle g}$  and ${\textstyle h}$  then ${\textstyle g}$  or ${\textstyle h}$  is an isomorphism.^[14]: 461

A proper minimal ring extension ${\textstyle T}$  of subring ${\textstyle R}$  occurs if the ring inclusion of ${\textstyle R}$  in to ${\textstyle T}$  is a minimal ring homomorphism. This implies the ring pair ${\textstyle (R,T)}$  has no proper intermediate ring.^[15]: 186

A minimal overring ${\textstyle T}$  of ring ${\textstyle R}$  occurs if ${\textstyle T}$  contains ${\textstyle R}$  as a subring, and the ring pair ${\textstyle (R,T)}$  has no proper intermediate ring.^[16]: 60

The Kaplansky ideal transform (Hayes transform, S-transform) of ideal ${\textstyle I}$  with respect to integral domain ${\textstyle R}$  is a subset of the fraction field ${\textstyle Q(R)}$ . This subset contains elements ${\textstyle x}$  such that for each element ${\textstyle y}$  of the ideal ${\textstyle I}$  there is a positive integer ${\textstyle n}$  with the product ${\textstyle x\cdot y^{n}}$  contained in integral domain ${\textstyle R}$ .^{[17][16]: 60}

Properties edit

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.^{[17][15]: 186}

The field of fractions of ${\textstyle R}$  contains minimal overring ${\textstyle T}$  of ${\textstyle R}$  when ${\textstyle R}$  is not a field.^[16]: 60

Assume an integrally closed integral domain ${\textstyle R}$  is not a field, If a minimal overring of integral domain ${\textstyle R}$  exists, this minimal overring occurs as the Kaplansky transform of a maximal ideal of ${\textstyle R}$ .^[16]: 60

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.