Overring

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

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

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

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

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

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.

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

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