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. Let Q ( A ) {\textstyle Q(A)} represent the field of fractions of an integral domain A {\textstyle A} . Ring B {\textstyle B} is an overring of integral domain A {\textstyle A} if A {\textstyle A} is a subring of B {\textstyle B} and B {\textstyle B} is a subring of the field of fractions Q ( A ) {\textstyle Q(A)} ; the relationship is A ⊆ B ⊆ Q ( A ) {\textstyle A\subseteq B\subseteq Q(A)} .
Properties
Ring of fractions The rings R A , S A , T A {\textstyle R_{A},S_{A},T_{A}} are the rings of fractions of rings R , S , T {\textstyle R,S,T} by multiplicative set A {\textstyle A} . Assume T {\textstyle T} is an overring of R {\textstyle R} and A {\textstyle A} is a multiplicative set in R {\textstyle R} . The ring T A {\textstyle T_{A}} is an overring of R A {\textstyle R_{A}} . The ring T A {\textstyle T_{A}} is the total ring of fractions of R A {\textstyle R_{A}} if every nonunit element of T A {\textstyle T_{A}} is a zero-divisor. Every overring of R A {\textstyle R_{A}} contained in T A {\textstyle T_{A}} is a ring S A {\textstyle S_{A}} , and S {\textstyle S} is an overring of R {\textstyle R} . Ring R A {\textstyle R_{A}} is integrally closed in T A {\textstyle T_{A}} if R {\textstyle R} is integrally closed in T {\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 R {\textstyle R} is locally nilpotentfree if every ring R M {\textstyle R_{M}} with maximal ideal M {\textstyle M} 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. These statements are equivalent for Noetherian ring R {\textstyle R} with integral closure R ¯ {\textstyle {\bar {R}}} .
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