Overring – Wikipedia
In mathematics, an overring of an integral domain contains the integral domain, and the overring itself is contained in a field called the field of fractions. Overrings provide an improved understanding of different types of rings and domains.
Definition[edit]
In this article, all rings are commutative rings, and ring and overring share the same identity element.
Let
represent the field of fractions of an integral domain
. Ring
is an overring of integral domain
if
is a subring of
and
is a subring of the field of fractions
;: 167 the relationship is
.: 373
Properties[edit]
Ring of fractions[edit]
The rings
are the rings of fractions of rings
by multiplicative set
.: 46 Assume
is an overring of
and
is a multiplicative set in
. The ring
is an overring of
. The ring
is the total ring of fractions of
if every nonunit element of
is a zero-divisor.: 52–53 Every overring of
contained in
is a ring
, and
is an overring of
.: 52–53 Ring
is integrally closed in
if
is integrally closed in
.: 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.: 199
An integral domain is a Dedekind domain if every ideal of the domain is a finite product of prime ideals.: 270
A ring’s restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element.: 52
A ring
is locally nilpotentfree if every ring
with maximal ideal
is free of nilpotent elements or a ring with every nonunit a zero divisor.: 52
An affine ring is the homomorphic image of a polynomial ring over a field.: 58
Properties[edit]
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.: 53
Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.: 53
These statements are equivalent for Noetherian ring
with integral closure
.: 57
These statements are equivalent for affine ring
with integral closure
.: 58
- Ring is locally nilpotentfree.
- Ring is a finite module.
- Ring is Noetherian.
An integrally closed local ring
is an integral domain or a ring whose non-unit elements are all zero-divisors.: 58
A Noetherian integral domain is a Dedekind ring if every overring of the Noetherian ring is integrally closed.: 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.: 200
Coherent rings[edit]
Definitions[edit]
A coherent ring is a commutative ring with each finitely generated ideal finitely presented.: 373 Noetherian domains and Prüfer domains are coherent.: 137
A pair
indicates a integral domain extension of
over
.: 331
Ring
is an intermediate domain for pair
if
is a subdomain of
and
is a subdomain of
.: 331
Properties[edit]
A Noetherian ring’s Krull dimension is 1 or less if every overring is coherent.: 373
For integral domain pair
,
is an overring of
if each intermediate integral domain is integrally closed in
.: 332 : 175
The integral closure of
is a Prüfer domain if each proper overring of
is coherent.: 137
The overrings of Prüfer domains and Krull 1-dimensional Noetherian domains are coherent.: 138
Prüfer domains[edit]
Properties[edit]
A ring has QR property if every overring is a localization with a multiplicative set.: 196 The QR domains are Prüfer domains.: 196 A Prüfer domain with a torsion Picard group is a QR domain.: 196 A Prüfer domain is a QR domain if the radical of every finitely generated ideal equals the radical generated by a principal ideal.: 500
The statement
is a Prüfer domain is equivalent to:: 56
The statement
is a Prüfer domain is equivalent to:: 167
- Each overring of is flat as a module.
- Each valuation overring of is a ring of fractions.
Minimal overring[edit]
Definitions[edit]
A minimal ring homomorphism
is an injective non-surjective homomorophism, and if the homomorphism
is a composition of homomorphisms
and
then
or
is an isomorphism.[14]: 461
A proper minimal ring extension
of subring
occurs if the ring inclusion of
in to
is a minimal ring homomorphism. This implies the ring pair
has no proper intermediate ring.: 186
A minimal overring
of ring
occurs if
contains
as a subring, and the ring pair
has no proper intermediate ring.: 60
The Kaplansky ideal transform (Hayes transform, S-transform) of ideal
with respect to integral domain
is a subset of the fraction field
. This subset contains elements
such that for each element
of the ideal
there is a positive integer
with the product
contained in integral domain
.: 60
Properties[edit]
Any domain generated from a minimal ring extension of domain
is an overring of
if
is not a field.: 186
The field of fractions of
contains minimal overring
of
when
is not a field.: 60
Assume an integrally closed integral domain
is not a field, If a minimal overring of integral domain
exists, this minimal overring occurs as the Kaplansky transform of a maximal ideal of
.: 60
Examples[edit]
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.: 168
The integer ring is a Prüfer ring, and all overrings are rings of quotients.: 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.
References[edit]
- Atiyah, Michael Francis; Macdonald, Ian G. (1969). Introduction to commutative algebra. Reading, Mass.: Addison-Wesley Publishing Company. ISBN 9780201407518.
- 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.
- Davis, Edward D (1962). “Overrings of commutative rings. I. Noetherian overrings” (PDF). Transactions of the American Mathematical Society. 104 (1): 52–61.
- Davis, Edward D (1964). “Overrings of commutative rings. II. Integrally closed overrings” (PDF). Transactions of the American Mathematical Society. 110 (2): 196–212.
- Davis, Edward D. (1973). “Overrings of commutative rings. III. Normal pairs” (PDF). Transactions of the American Mathematical Society: 175–185.
- Dobbs, David E.; Shapiro, Jay (2006). “A classification of the minimal ring extensions of an integral domain” (PDF). Journal of Algebra. 305 (1): 185–193. doi:10.1016/j.jalgebra.2005.10.005.
- Dobbs, David E.; Shapiro, Jay (2007). “Descent of minimal overrings of integrally closed domains to fixed rings”. Houston Journal of Mathematics. 33 (1).
- Ferrand, Daniel; Olivier, Jean-Pierre (1970). “Homomorphismes minimaux d’anneaux” (PDF). Journal of Algebra. 16 (3): 461–471.
- Fontana, Marco; Papick, Ira J. (2002), “Dedekind and Prüfer domains”, in Mikhalev, Alexander V.; Pilz, Günter F. (eds.), The concise handbook of algebra, Kluwer Academic Publishers, Dordrecht, pp. 165–168, ISBN 9780792370727
- Fuchs, Laszlo; Heinzer, William; Olberding, Bruce (2004), “Maximal prime divisors in arithmetical rings”, Rings, modules, algebras, and abelian groups, Lecture Notes in Pure and Appl. Math., vol. 236, Dekker, New York, pp. 189–203, MR 2050712
- Lane, Saunders Mac; Schilling, O. F. G. (1939). “Infinite number fields with Noether ideal theories”. American Journal of Mathematics. 61 (3): 771–782.
- Papick, Ira J. (1978). “A Remark on Coherent Overrings” (PDF). Canad. Math. Bull. 21 (3): 373–375.
- Papick, Ira J. (1979). “Coherent overrings” (PDF). Canadian Mathematical Bulletin. 22 (3): 331–337.
- Papick, Ira J. (1980). “A note on proper overrings”. Rikkyo Daigaku sugaku zasshi. 28 (2): 137–140.
- Pendleton, Robert L. (1966). “A characterization of Q-domains”. Bull. Amer. Math. Soc. 72 (4): 499–500.
- Sato, Junro; Sugatani, Takasi; Yoshida, Ken-ichi (January 1992). “On minimal overrings of a noetherian domain”. Communications in Algebra. 20 (6): 1735–1746. doi:10.1080/00927879208824427.
- Zariski, Oscar; Samuel, Pierre (1965). Commutative algebra. New York: Springer-Verlag. ISBN 978-0-387-90089-6.
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