VALUATIONS OF k((X1,…,Xn)) WITH PREASSIGNED GROUP OF VALUES

2004 ◽  
Vol 03 (04) ◽  
pp. 453-468 ◽  
Author(s):  
A. GRANJA
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Algebraic Extension ◽  
Power Series Ring ◽  
Quotient Field ◽  
Series Ring ◽  
Rational Rank

Let Γ be a totally ordered abelian group of finite rational rank r. Consider s and n two non-negative integers with r+s≤n and denote by K a field. The main purpose of this paper is to construct a s-dimensional valuation v of the quotient field K((X1,…,Xn)) of the corresponding power series ring K[[X1,…,Xn]] such that v birrationally dominates K[[X1,…,Xn]] and has Γ as group of values. This is possible except for the case in which Γ=ℤ is the group of the integers, s=0, n≥2 and K does not admit an infinite algebraic extension. In this last case, we show that there does not exist such a valuation.

scholarly journals When is a power series ring n-root closed?

1997 ◽  
Vol 114 (2) ◽  
pp. 111-131 ◽  
Author(s):  
David F. Anderson ◽  
David E. Dobbs ◽  
Moshe Roitman
Keyword(s):  
Power Series ◽  
Power Series Ring ◽  
Series Ring

scholarly journals Completions of rings of invariants and their divisor class groups

1978 ◽  
Vol 72 ◽  
pp. 71-82 ◽  
Author(s):  
Phillip Griffith
Keyword(s):  
Power Series ◽  
Maximal Ideal ◽  
Total Degree ◽  
Class Groups ◽  
Divisor Class ◽  
Power Series Ring ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Ring Of Polynomials ◽  
Formal Power

Let k be a field and let A = be a normal graded subring of the full ring of polynomials R = k[X1, · · ·, Xn] (where R always is graded via total degree and A0 = k). R. Fossum and the author [F-G] observed that the completion  at the irrelevant maximal ideal of A is isomorphic to the subring of the formal power series ring R̂ = k[[X1, · ·., Xn]] and, moreover, that  is a ring of invariants of an algebraic group whenever A is.

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.

Polynomial and power series ring extensions from sequences

2020 ◽  
pp. 2250048
Author(s):  
Gyu Whan Chang ◽  
Phan Thanh Toan
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Trivial Case ◽  
Addition Formula ◽  
Power Series Ring ◽  
Series Ring ◽  
Ring Extensions ◽  
General Sequences ◽  
Divisibility Properties ◽  
Positive Integers

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

Graded Complexes Over Power Series Rings

1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts
Keyword(s):  
Power Series ◽  
Local Ring ◽  
Simple Structure ◽  
Free Resolution ◽  
Koszul Complex ◽  
Local Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Noetherian Local Ring ◽  
Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

On semilocal, Bézout and distributive generalized power series rings

2015 ◽  
Vol 25 (05) ◽  
pp. 725-744 ◽  
Author(s):  
Ryszard Mazurek ◽  
Michał Ziembowski
Keyword(s):  
Power Series ◽  
Neumann Series ◽  
Polynomial Rings ◽  
Group Rings ◽  
Power Series Ring ◽  
Generalized Power Series ◽  
Series Ring ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Totally Ordered Monoid

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

scholarly journals PF-rings of generalised power series

1998 ◽  
Vol 57 (3) ◽  
pp. 427-432 ◽  
Author(s):  
Zhongkui Liu
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Power Series Ring ◽  
Series Ring ◽  
Ordered Monoid

Let R be a commutative ring and (S, ≤) a strictly ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. We show that the generalised power series ring [[RS ≤]] is a PF-ring if and only if R is a PF-ring.

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