Warfield Duality and Module Extensions Over a Noetherian Domain

1995 ◽  
pp. 239-248 ◽  
Author(s):  
H. Pat Goeters
Keyword(s):  
Noetherian Domain

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.

Discrete valuation overrings of a graded Noetherian domain

2018 ◽  
Vol 10 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Gyu Whan Chang ◽  
Dong Yeol Oh
Keyword(s):  
Noetherian Domain

scholarly journals ON SEPARABLE AND -FORMS

2018 ◽  
Vol 239 ◽  
pp. 346-354
Author(s):  
AMARTYA KUMAR DUTTA ◽  
NEENA GUPTA ◽  
ANIMESH LAHIRI
Keyword(s):  
Characteristic Zero ◽  
One Dimensional ◽  
Noetherian Domain ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 337-353 ◽  
Author(s):  
MAHMOOD BEHBOODI
Keyword(s):  
Prime Ring ◽  
Artinian Ring ◽  
Prime Radical ◽  
Nilpotent Elements ◽  
Noetherian Domain ◽  
Prime Submodules ◽  
Strongly Nilpotent ◽  
Commutative Domain ◽  
Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

The Width of a Module

1970 ◽  
Vol 22 (1) ◽  
pp. 102-115 ◽  
Author(s):  
Michael Wichman
Keyword(s):  
Finite Rank ◽  
Krull Dimension ◽  
Minimum Condition ◽  
Finite Width ◽  
Noetherian Domain ◽  
Dimension One

An R-module N is said to have finite width n if n is the smallest integer such that for any set of n + 1 elements of N, at least one of the elements is in the submodule generated by the remaining n. The width of N over R will be denoted by W(R, N).The notion of width was introduced by Brameret [2, p. 3605]. However, Cohen [3] investigated rings of finite rank, which, in the case that R is a local Noetherian domain, is equivalent to width (Proposition 1.6). He showed that finite width of R was both equivalent to R having Krull dimension one, and to R having the restricted minimum condition (Theorem 1.12).

Noetherian domains in which every ideal is isomorphic to a trace ideal

2019 ◽  
Vol 19 (10) ◽  
pp. 2050200
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Large Family ◽  
One Dimensional ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Trace Ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

scholarly journals Formal fibers and birational extensions

1993 ◽  
Vol 131 ◽  
pp. 1-38 ◽  
Author(s):  
William Heinzer ◽  
Christel Rotthaus ◽  
Judith D. Sally
Keyword(s):  
Noetherian Ring ◽  
Important Condition ◽  
Quotient Field ◽  
Noetherian Domain ◽  
Definition Of ◽  
Adic Completion

Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Localization in enveloping algebras

1983 ◽  
Vol 93 (3) ◽  
pp. 467-475 ◽  
Author(s):  
A. I. Lichtman
Keyword(s):  
Lie Algebra ◽  
Skew Field ◽  
Enveloping Algebras ◽  
Finite Dimensional ◽  
Noetherian Domain ◽  
Term Field

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.

Down–up algebras over a polynomial base ring 𝕂[t1,…,tn]

2015 ◽  
Vol 15 (02) ◽  
pp. 1650024
Author(s):  
Xin Tang
Keyword(s):  
Prime Ring ◽  
Global Dimension ◽  
Noetherian Domain ◽  
Base Ring ◽  
Kirillov Dimension ◽  
Krull Dimensions

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t1,…,tn] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).

scholarly journals A bound on certain local cohomology modules and application to ample divisors

2001 ◽  
Vol 163 ◽  
pp. 87-106 ◽  
Author(s):  
Claudia Albertini ◽  
Markus Brodmann
Keyword(s):  
Local Cohomology ◽  
Positive Characteristic ◽  
Local Cohomology Module ◽  
Perfect Field ◽  
Noetherian Domain ◽  
Local Cohomology Modules ◽  
Generic Fibre ◽  
Cohomology Modules ◽  
Ample Divisors ◽  
Torsion Free

We consider a positively graded noetherian domain R = ⊕n∈NoRn for which R0 is essentially of finite type over a perfect field K of positive characteristic and we assume that the generic fibre of the natural morphism π: Y = Proj(R) → Y0 = Spec(R0) is geometrically connected, geometrically normal and of dimension > 1. Then we give bounds on the “ranks” of the n-th homogeneous part H2(R)n of the second local cohomology module of R with respect to R+:= ⊕m>0Rm for n < 0. If Y is in addition normal, we shall see that the R0-modules H2R+ (R)n are torsion-free for all n < 0 and in this case our bounds on the ranks furnish a vanishing result. From these results we get bounds on the first cohomology of ample invertible sheaves in positive characteristic.

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