scholarly journals Groups generated by elements with rational fixed points

1997 ◽  
Vol 40 (1) ◽  
pp. 19-30 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Fixed Points ◽  
Large Class ◽  
Integral Domain ◽  
Structure Theorem ◽  
Quotient Field ◽  
Linear Fractional Transformations ◽  
Krull Domain ◽  
Dedekind Domains ◽  
Precise Version

Let R be a commutative integral domain and let S be its quotient field. The group GL2(R) acts on Ŝ = S ∪ {∞} as a group of linear fractional transformations in the usual way. Let F2(R, z) be the stabilizer of z ∈ Ŝ in GL2(R) and let F2(R) be the subgroup generated by all F2(R, z). Among the subgroups contained in F2(R) are U2(R), the subgroup generated by all unipotent matrices, and NE2(R), the normal subgroup generated by all elementary matrices.We prove a structure theorem for F2(R, z), when R is a Krull domain. A more precise version holds when R is a Dedekind domain. For a large class of arithmetic Dedekind domains it is known that the groups NE2(R),U2(R) and SL2(R) coincide. An example is given for which all these subgroups are distinct.

When is R[θ] integrally closed?

2016 ◽  
Vol 15 (05) ◽  
pp. 1650091 ◽  
Author(s):  
Sudesh K. Khanduja ◽  
Bablesh Jhorar
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Minimal Polynomial ◽  
Closed Domain ◽  
Necessary And Sufficient Condition ◽  
Quotient Field ◽  
Maximal Ideals ◽  
Dedekind Domains ◽  
Integrally Closed ◽  
Necessary And Sufficient

Let [Formula: see text] be an integrally closed domain with quotient field [Formula: see text] and [Formula: see text] be an element of an integral domain containing [Formula: see text] with [Formula: see text] integral over [Formula: see text]. Let [Formula: see text] be the minimal polynomial of [Formula: see text] over [Formula: see text] and [Formula: see text] be a maximal ideal of [Formula: see text]. Kummer proved that if [Formula: see text] is an integrally closed domain, then the maximal ideals of [Formula: see text] which lie over [Formula: see text] can be explicitly determined from the irreducible factors of [Formula: see text] modulo [Formula: see text]. In 1878, Dedekind gave a criterion known as Dedekind Criterion to be satisfied by [Formula: see text] for [Formula: see text] to be integrally closed in case [Formula: see text] is the localization [Formula: see text] of [Formula: see text] at a nonzero prime ideal [Formula: see text] of [Formula: see text]. Indeed he proved that if [Formula: see text] is the factorization of [Formula: see text] into irreducible polynomials modulo [Formula: see text] with [Formula: see text] monic, then [Formula: see text] is integrally closed if and only if for each [Formula: see text], either [Formula: see text] or [Formula: see text] does not divide [Formula: see text] modulo [Formula: see text], where [Formula: see text]. In 2006, a similar necessary and sufficient condition was given by Ershov for [Formula: see text] to be integrally closed when [Formula: see text] is the valuation ring of a Krull valuation of arbitrary rank (see [Comm. Algebra. 38 (2010) 684–696]). In this paper, we deal with the above problem for more general rings besides giving some equivalent versions of Dedekind Criterion. The well-known result of Uchida in this direction proved for Dedekind domains has also been deduced (cf. [Osaka J. Math. 14 (1977) 155–157]).

Integral Domains Whose w-Integral Closures Are Krull Domains

2020 ◽  
Vol 27 (02) ◽  
pp. 287-298
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Krull Domain ◽  
Integral Closures ◽  
Krull Domains

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

Some Open Questions on Minimal Primes of a Krull Domain

1968 ◽  
Vol 20 ◽  
pp. 1261-1264 ◽  
Author(s):  
Paul M. Eakin ◽  
William J. Heinzer
Keyword(s):  
Integral Domain ◽  
Zero Element ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Krull Domain ◽  
Open Questions ◽  
Valuation Rings ◽  
Rank One

Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set {Vα} of rank one discrete valuation rings such that A = ∩αVα and such that each non-zero element of A is a non-unit in only finitely many of the Vα. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104).

scholarly journals Hecke invariants of knot groups

1974 ◽  
Vol 15 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Integral Domain ◽  
Unique Fixed Point ◽  
Projective Line ◽  
Transcendence Degree ◽  
Quotient Field ◽  
Prime Field ◽  
Projective Transformations ◽  
The Right

For each characteristic p, let Fp be the prime field and let Ώp be a fixed universal field which is algebraically closed and of infinite transcendence degree over Fp. When p = 0 we take Ώp = ℂ. Let F be a subfield of Ώp and let R be an integral domain whose quotient field is F. We abbreviate SL(2, R), PGL(2, R), PSL(2, R) to SL(R), PGL(R), PSL(R) respectively, and we cohsider PSL(R) as a group of projective transformations of the projective line ℘(Ώp) and of the “subline” ℘(F) ⊂ ℘(ΏP). The elements of PSL(R) are classified by the number of fixed points they have on ℘(F). If x ∈ PSL(R) has one such fixed point P, then P is the unique fixed point of x on ℘(ΏP) and x is called parabolic. All other x (except the identity E) have two distinct fixed points on ℘(Ώp) and x is called hyperbolic if these are on ℘(F), and elliptic otherwise. We put symbols for operators on the right.

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]).

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

On Representations of Orders Over Dedekind Domains

1960 ◽  
Vol 12 ◽  
pp. 107-125 ◽  
Author(s):  
D. G. Higman
Keyword(s):  
Integral Domain ◽  
Homological Algebra ◽  
Dedekind Domain ◽  
Main Concern ◽  
Direct Sums ◽  
Dedekind Domains ◽  
The Ideal

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

Convex Directed Subgroups of a Group of Divisibility

1974 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Integral Domain ◽  
Multiplicative Group ◽  
Ordered Group ◽  
Quotient Field ◽  
Partially Ordered ◽  
Group Of Units ◽  
Partially Ordered Group ◽  
Group Of Divisibility

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

Factoring Ideals into Semiprime Ideals

1978 ◽  
Vol 30 (6) ◽  
pp. 1313-1318 ◽  
Author(s):  
N. H. Vaughan ◽  
R. W. Yeagy
Keyword(s):  
Integral Domain ◽  
Dedekind Domain ◽  
Prime Ideals ◽  
Semiprime Ideal ◽  
Dedekind Domains

Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

scholarly journals On complete integral closure and Archimedean valuation domains

1996 ◽  
Vol 61 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Extension Field ◽  
Quotient Field ◽  
Valuation Domains ◽  
Complete Integral Closure

AbstractSuppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

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