scholarly journals The Homological Dimension of a Quotient Field

1995 ◽  
pp. 197-201
Author(s):  
Irving Kaplansky
Keyword(s):  
Homological Dimension ◽  
Quotient Field

CROSSED PRODUCT ORDERS OVER VALUATION RINGS

2001 ◽  
Vol 33 (5) ◽  
pp. 520-526 ◽  
Author(s):  
JOHN S. KAUTA
Keyword(s):  
Crossed Product ◽  
Homological Dimension ◽  
Crossed Products ◽  
Quotient Field ◽  
Group Of Units ◽  
Valuation Rings ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Product Order ◽  
Natural Way

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

scholarly journals The Homological Dimension of a Quotient Field

1966 ◽  
Vol 27 (1) ◽  
pp. 139-142 ◽  
Author(s):  
Irving Kaplansky
Keyword(s):  
Integral Domain ◽  
Projective Resolution ◽  
Minimal Length ◽  
Homological Dimension ◽  
Quotient Field

Let R be an integral domain, Q its quotient field. For any R-module A we write d(A) for its homological dimension (minimal length of a projective resolution), or dR(A) if it is necessary to call attention to R.

scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

scholarly journals Homological dimension in semigroup algebras

1971 ◽  
Vol 18 (3) ◽  
pp. 404-413 ◽  
Author(s):  
William R Nico
Keyword(s):  
Homological Dimension ◽  
Semigroup Algebras

Intersections of quotient rings and Prüfer v-multiplication domains

2016 ◽  
Vol 15 (08) ◽  
pp. 1650149 ◽  
Author(s):  
Said El Baghdadi ◽  
Marco Fontana ◽  
Muhammad Zafrullah
Keyword(s):  
Integral Domain ◽  
Algebraic Approach ◽  
Quotient Field ◽  
Topological Characterization ◽  
Locally Finite ◽  
Star Operation ◽  
Star Operations ◽  
Quotient Rings ◽  
Special Case

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call an overring [Formula: see text] of [Formula: see text] a subring of [Formula: see text] containing [Formula: see text] as a subring. A family [Formula: see text] of overrings of [Formula: see text] is called a defining family of [Formula: see text], if [Formula: see text]. Call an overring [Formula: see text] a sublocalization of [Formula: see text], if [Formula: see text] has a defining family consisting of rings of fractions of [Formula: see text]. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535–2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer [Formula: see text]-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer [Formula: see text]-multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d’anneaux intègers, J. Algebra 43 (1976) 55–60] and Proposition 3.2 of [N. Dessagnes, Intersections d’anneaux de Mori — exemples, Port. Math. 44 (1987) 379–392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P vMDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer [Formula: see text]-multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]).

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

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