Uniserial Dimensions of Finitely Generated Primary Modules Over A Discrete Valuation Domain

AbstractLet D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D, set It = ∪{JV = (J-1)-1 | J is a nonzero finitely generated subideal of I}; this defines the t-operation, a particularly useful star-operation on D. We discuss the t-operation on R[[X]], paying particular attention to the relation between t- dim(R) and t- dim(R[[X]]). We show that if P is a t-prime of R, then P[[X]] contains a t-prime which contracts to P in R, and we note that this does not quite suffice to show that t- dim(R[[X]]) ≥ t- dim(R) in general. If R is Noetherian, it is easy to see that t- dim(R[[X]]) ≥ t- dim(R), and we show that we have equality in the case of t-dimension 1. We also observe that if V is a valuation domain, then t-dim(V[[X]]) ≥ t- dim(V), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M, then MV[[X]] is a maximal t-ideal of V[[X]].

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On the Factorization of Polynomials Over Discrete Valuation Domains

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2014-0023 ◽

2014 ◽

Vol 22 (1) ◽

pp. 273-280

Author(s):

Doru Ştefănescu

Keyword(s):

Valuation Domain ◽

Discrete Valuation Domain ◽

Valuation Domains ◽

Univariate Polynomials ◽

Factorization Of Polynomials

AbstractWe study some factorization properties for univariate polynomials with coefficients in a discrete valuation domain (A,v). We use some properties of the Newton index of a polynomial to deduce conditions on v(ai) that allow us to find some information on the degree of the factors of F.

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Computing the V-saturation of finitely-generated submodules of V[X]m where V is a valuation domain

Journal of Symbolic Computation ◽

10.1016/j.jsc.2015.02.006 ◽

2016 ◽

Vol 72 ◽

pp. 196-205 ◽

Cited By ~ 4

Author(s):

Lionel Ducos ◽

Samiha Monceur ◽

Ihsen Yengui

Keyword(s):

Valuation Domain ◽

Finitely Generated

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Mixed modules of finite torsion-free rank over a discrete valuation domain

Forum Mathematicum ◽

10.1515/forum.2006.004 ◽

2006 ◽

Vol 18 (1) ◽

Author(s):

David M Arnold ◽

K. M Rangaswamy

Keyword(s):

Valuation Domain ◽

Discrete Valuation Domain ◽

Free Rank ◽

Torsion Free Rank ◽

Torsion Free

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Stacked submodules of torsion modules over discrete valuation domains

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700037849 ◽

2003 ◽

Vol 68 (3) ◽

pp. 439-447 ◽

Cited By ~ 2

Author(s):

Pudji Astuti ◽

Harald K. Wimmer

Keyword(s):

Valuation Domain ◽

Torsion Module ◽

Discrete Valuation Domain ◽

Valuation Domains ◽

Torsion Modules

A submodule W of a torsion module M over a discrete valuation domain is called stacked in M if there exists a basis ℬ of M such that multiples of elements of ℬ form a basis of W. We characterise those submodules which are stacked in a pure submodule of M.

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