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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
editRing is an overring of ring if is a subring of and is a subring of the total ring of fractions ; the relationship is .[1]: 167
Properties
editUnless otherwise stated, all rings are commutative rings, and each ring and its overring share the same identity element.
Ring of fractions
editDefinitions
editThe ring is the ring of fractions (ring of quotients, localization) of ring by multiplicative system set , .[2]: 46
Theorems
editAssume is an overring of and is a multiplicative system and . The implications are:[3]: 52–53
- The ring is an overring of . The ring is the total ring of fractions of if every nonunit element of is a zero-divisor.
- Every overring of contained in is a ring , and is an overring of .
- Ring is integrally closed in if is integrally closed in .
Noetherian domain
editDefinitions
editA 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.[2]: 199
An integral domain is a Dedekind domain if every ideal of the domain is a finite product of prime ideals.[2]: 270
A ring's restricted dimension is the maximum rank among the ranks of all prime ideals that contain a regular element.[3]: 52
A ring is locally nilpotentfree if every , generated by each maximal ideal , is free of nilpotent elements or a ring with every non-unit a zero divisor.[3]: 52
An affine ring is the homomorphic image of a polynomial ring over a field.[3]: 58
The torsion class group of a Dedekind domain is the group of fractional domains modulo the principal fractional ideals subgroup.[4]: 96 [5]: 200
Theorems
editEvery overring of a Dedekind ring is a Dedekind ring.[6][7]
Every overrring of a Direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.[3]: 53
Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.[3]: 53
These statements are equivalent for Noetherian ring with integral closure .[3]: 57
- Every overring of is a Noetherian ring.
- For each maximal ideal of , every overring of is a Noetherian ring.
- Ring is locally nilpotentfree with restricted dimension 1 or less.
- Ring is Noetherian, and ring has restricted dimension 1 or less.
- Every overring of is integrally closed.
These statements are equivalent for affine ring with integral closure .[3]: 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.[3]: 58
A Noetherian integral domain is a Dedekind ring if and only if every overring of the Noetherian ring is integrally closed.[5]: 198
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.[5]: 200
Coherent rings
editDefinitions
editA coherent ring is a commutative ring with each finitely generated ideal finitely presented.[8]: 373 Noetherian domains and Prüfer domains are coherent.[9]: 137
A pair indicates that is an integral domain extension over with .[10]: 331
An intermediate domain for pair indicates this relationship .[10]: 331
Theorems
editA Noetherian ring's Krull dimension is 1 or less if every overring is coherent.[8]: 373
For integral domain pair , is an overring of if each intermediate integral domain is integrally closed in .[10]: 332 [11]: 175
The integral closure of is a Prüfer domain if each proper overring of is coherent.[9]: 137
The overrings of Prüfer domains and Krull 1-dimensional Noetherian domains are coherent.[9]: 138
Prüfer domains
editTheorems
editA ring has QR property if every overring is a localization with a multiplicative system.[12]: 196
- QR domains are Prüfer domains.[12]: 196
- A Prüfer domain with a torsion Picard group is a QR domain.[12]: 196
- 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.[13]: 500
The statement is a Prüfer domain is equivalent to:[14]: 56
- Each overring of is the intersection of localizations of , and is integrally closed.
- Each overring of is the intersection of rings of fractions of , and is integrally closed.
- Each overring of has prime ideals that are extensions of the prime ideals of , and is integrally closed.
- Each overring of has at most 1 prime ideal lying over any prime ideal of , and is integrally closed
- Each overring of is integrally closed.
- Each overring of is coherent.
The statement is a Prüfer domain is equivalent to:[1]: 167
- Each overring of is flat as a module.
- Each valuation overring of is a ring of fractions.
Minimal overring
editDefinitions
editA minimal ring homomorphism is an injective non-surjective homomorophism, and any decomposition implies or is an isomorphism.[15]: 461
A proper minimal ring extension of subring occurs when the ring inclusion is a minimal ring homomorphism. This implies the ring pair has no proper intermediate ring.[16]: 186
A minimal overring integral domain of integral domain occurs when contains as a subring, and the ring pair has no proper intermediate ring.[17]: 60
The Kaplansky ideal transform (Hayes transform, S-transform) for ideal in ring is:[18][17]: 60
Theorems
editAny domain generated from a minimal ring extension of domain is an overring of if is not a field.[18][16]: 186 The 1st of 3 types of minimal ring extensions of domain generates a domain and minimal overring of that contains .[16]: 191
The field of fractions of contains minimal overring of when is not a field.[17]: 60
If a minimal overring of a non-field integrally closed integral domain exists, this minimal overring occurs as the Kaplansky transform of a maximal ideal of .[17]: 60
Examples
editThe 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.[5]: 196 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.
Notes
edit- ^ a b c Fontana & Papick 2002.
- ^ a b c Zariski & Samuel 1965.
- ^ a b c d e f g h i Davis 1962.
- ^ Atiyah & Macdonald 1969.
- ^ a b c d Davis 1964.
- ^ Cohen 1950.
- ^ Lane & Schilling 1939.
- ^ a b Papick 1978.
- ^ a b c Papick 1980.
- ^ a b c Papick 1979.
- ^ Davis 1973.
- ^ a b c Fuchs, Heinzer & Olberding 2004.
- ^ Pendleton 1966.
- ^ Bazzoni & Glaz 2006.
- ^ Ferrand & Olivier 1970.
- ^ a b c Dobbs & Shapiro 2006.
- ^ a b c d Dobbs & Shapiro 2007.
- ^ a b Sato, Sugatani & Yoshida 1992.
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" (PDF). ouston Journal of Mathematics. 33 (1).
- Ferrand, Daniel; Olivier, Jean-Pierre (1970). "Homomorphismes minimaux d'anneaux". 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" (PDF). 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.
Related categories
editCategory:Ring theory Category:Algebraic structures Category:Commutative algebra