scholarly journals A Note on Quotient Fields of Power Series Rings

1994 ◽  
Vol 37 (2) ◽  
pp. 162-164 ◽  
Author(s):  
Huah Chu ◽  
Yi-Chuan Lang
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Algebraic Degree ◽  
Closed Domain ◽  
Quotient Field ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Power Series Rings

AbstractLet R be an integral domain with quotient field K. If R has an overling S ≠ K, such that S[X] is integrally closed, then the "algebraic degree" of K((X)) over the quotient field of R[X] is infinite. In particular, it holds for completely integrally closed domain or Noetherian domain R.

Condensed Domains

2003 ◽  
Vol 46 (1) ◽  
pp. 3-13 ◽  
Author(s):  
D. D. Anderson ◽  
Tiberiu Dumitrescu
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Field Extension ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Closed Domain ◽  
Local Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Finite Integral

AbstractAn integral domain D with identity is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, IJ = {ij ; i ∈ I; j ∈ J} (resp., IJ = iJ for some i ∈ I or IJ = Ij for some j ∈ J). We show that for a Noetherian domain D, D is condensed if and only if Pic(D) = 0 and D is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain D is strongly condensed if and only if D is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension k ⊆ K, the domain D = k + XK[[X]] is condensed if and only if [K : k] ≤ 2 or [K : k] = 3 and each degree-two polynomial in k[X] splits over k, while D is strongly condensed if and only if [K : k] ≤ 2.

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]).

On S-GCD domains

2019 ◽  
Vol 18 (04) ◽  
pp. 1950067 ◽  
Author(s):  
D. D. Anderson ◽  
Ahmed Hamed ◽  
Muhammad Zafrullah
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Integral Domain ◽  
Class Group ◽  
Nonzero Ideal ◽  
Noetherian Domain ◽  
Power Series Rings ◽  
Formal Power ◽  
Equivalent Conditions ◽  
Unit Formula

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

Weak Normalization of Power Series Rings

1995 ◽  
Vol 38 (4) ◽  
pp. 429-433 ◽  
Author(s):  
David E. Dobbs ◽  
Moshe Roitman
Keyword(s):  
Power Series ◽  
Integral Domain ◽  
Power Series Ring ◽  
Series Ring ◽  
Power Series Rings

AbstractIt is proved that if r* is the weak normalization of an integral domain r, then the weak normalization of the power series ring r[[x1,....xn]] is contained in R*[[X1,....Xn]]. Consequently, if R is a weakly normal integral domain, then R[[X1,....Xn]] is also weakly normal.

An answer to a problem about the number of overrings

2016 ◽  
Vol 15 (06) ◽  
pp. 1650022 ◽  
Author(s):  
M. Ben Nasr
Keyword(s):  
Open Problem ◽  
Integral Domain ◽  
Integral Closure ◽  
Closed Domain ◽  
Prüfer Domain ◽  
Finite Spectrum ◽  
Integrally Closed

Let [Formula: see text] be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure [Formula: see text] is a Prüfer domain with finite spectrum and there are only finitely many rings between [Formula: see text] and [Formula: see text]. Jaballah solved the problem of counting the overrings in the case [Formula: see text] but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math. 23 (2005) 353–360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.

scholarly journals Note on U-closed semigroup rings

1999 ◽  
Vol 59 (3) ◽  
pp. 467-471 ◽  
Author(s):  
Ryûki Matsuda
Keyword(s):  
Abelian Group ◽  
Integral Domain ◽  
Free Abelian Group ◽  
Semigroup Ring ◽  
Closed Domain ◽  
Semigroup Rings ◽  
Quotient Field ◽  
Torsion Free Abelian Group ◽  
Torsion Free

Let D be an integral domain with quotient field K. If α2 − α ∈ D and α3 − α2 ∈ D imply α ∈ D for all elements α of K, then D is called a u-closed domain. A submonoid S of a torsion-free Abelian group is called a grading monoid. We consider the semigroup ring D[S] of a grading monoid S over a domain D. The main aim of this note is to determine conditions for D[S] to be u-closed. We shall show the following Theorem: D[S] is u-closed if and only if D is u-closed.

On a Condition of J. Ohm for Integral Domains

1968 ◽  
Vol 20 ◽  
pp. 970-983 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Closed Domain ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Integrally Closed

This paper originated mainly from results presented in a paper by J. Ohm (13), and, to a lesser degree, from results of Gilmer in (3). Ohm's paper is concerned with the validity of the equation (x, y)n = (xn, yn) for each pair of elements x, y of an integral domain D with identity. If D is a Prüfer domain, the above equation is valid for all x, y ϵ D (7, p. 244). Butts and Smith have shown (2) that if (x, y)2 = (x2, y2) for all x, y of the integrally closed domain D, then D is a Prüfer domain.

Prüfer v-multiplication domains and the completeness of ω-ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 446-453
Author(s):  
Youhua Chen ◽  
Fanggui Wang ◽  
Huayu Yin
Keyword(s):  
Prime Ideal ◽  
Closed Domain ◽  
Quotient Field ◽  
Integrally Closed

Let R be a domain with quotient field K. It is proved that R is an integrally closed domain if and only if every nonzero t-ideal of R is complete, if and only if every nonzero v-ideal of R is complete. We also obtain that every prime ideal of an integrally closed domain is integrally closed, and every strongly prime ideal of a domain is integrally closed. Moreover, we introduce the notion of w-cancellation ideals and give some equivalent characterizations of PVMDs. In particular, it is proved that R is a PVMD if and only if every w-ideal of R is complete.

CONSTRUCTING CHAINS OF PRIMES IN POWER SERIES RINGS, II

2012 ◽  
Vol 12 (01) ◽  
pp. 1250123 ◽  
Author(s):  
K. ALAN LOPER ◽  
THOMAS G. LUCAS
Keyword(s):  
Lower Bound ◽  
Power Series ◽  
Polynomial Ring ◽  
Integral Domain ◽  
Valuation Domain ◽  
Main Concern ◽  
Power Series Ring ◽  
One Dimensional ◽  
Series Ring ◽  
Power Series Rings

For an integral domain D of dimension n, the dimension of the polynomial ring D[ x ] is known to be bounded by n + 1 and 2n + 1. While n + 1 is a lower bound for the dimension of the power series ring D[[ x ]], it often happens that D[[ x ]] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a one-dimensional nondiscrete valuation domain. The main concern here is in developing a scheme by which such chains can be constructed in the gap between MV[[ x ]] and M[[ x ]] when V is a one-dimensional nondiscrete valuation domain with maximal ideal M. A consequence of these constructions is that there are chains of primes similar to the set of ω1 transfinite sequences of 0's and 1's ordered lexicographically.

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