Valuation Derived from Graded Ring and Module and Krull Dimension Properties (original) (raw)
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On the structure of the graded algebra associated to a valuation
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The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra gr v (R) of a subring (R, m) of a valuation ring Ov, for which Kv := Ov/mv = R/m, is isomorphic to Kv t v(R) , where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist.
Journal of the Mathematical Society of Japan, 1978
In this paper, we study a Noetherian graded ring RRR and the category of graded R-modules. We consider injective objects of this category and we define the graded Cousin complex of a graded R-module MMM . These concepts are essential in this paper
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1997
The graded algebra g%R relative to a valuation v of the quotient field of a noetherian local domain R centered at R is an S-graded K-algebra where S is the value semigroup of v and K the residue field of R. We show how the semigroup theoretical properties of S allow to describe a minimal system of homogeneous generators for grvR and to obtain an S-graded minimal resolution of grvR as A[v]-module, A[v] being certain associated polynomial ring. We derive a formula to obtain the number of generators of a fixed degree (in S) for each syzygy module and so to compute them in combinatorial terms.
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Journal of Pure and Applied Algebra, 2023
In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn in terms of the key polynomials that define Qn. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
Proceedings of the American Mathematical Society, 1989
Let R be a faithfully S-graded ring, where 5 is a submoniod of a torsion-free commutative group and S has no nontrivial units. In case R is a prime Krull order we give necessary and sufficient conditions for R to be a crossed product (respectively a polynomial ring).
The Class of Noetherian Rings With Finite Valuation Dimension
Mathematics and Statistics, 2021
Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They’ve shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valuation dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian.
On the global balanced-projective dimension of valuation domains
Periodica Mathematica Hungarica - PERIOD MATH HUNG, 2003
Assuming the General Continuum Hypothesis, the global balanced-projective dimension of a valuation domain is determined. We show that it is always equal to the supremum of the projective dimensions of torsion-free modules.
Dimension of Crystalline Graded Rings
The global dimension of a ring governs many useful abilities. For example, it is semi-simple if the global dimension is 0, hereditary if it is 1 and so on. We will calculate the global dimension of a Crystalline Graded Ring, as defined in the paper by E. Nauwelaerts and F. Van Oystaeyen, Introducing Crystalline Graded Algebras, Algebras and Representation Theory vol 11(2008), no. 2, 133--148.. We will apply this to derive a condition for the Crystalline Graded Ring to be semiprime. In the last section, we give a little bit of attention to the Krull-dimension.
A generalization of the classical krull dimension for modules
2006
In this article, we introduce and study a generalization of the classical Krull dimension for a module R M. This is defined to be the length of the longest strong chain of prime submodules of M (defined later) and, denoted by Cl.K.dim(M). This notion is analogous to that of the usual classical Krull dimension of a ring. This dimension, Cl.K.dim(M) exists if and only if M has virtual acc on prime submodules; see Section 2. If R is a ring for which Cl.K.dim(R) exists, then for any left R-module M, Cl.K.dim(M) exists and is no larger than Cl.K.dim(R). Over any ring, all homogeneous semisimple modules and over a PI-ring (or an FBN-ring), all semisimple modules as well as, all Artinian modules with a prime submodule lie in the class of modules with classical Krull dimension zero. For a multiplication module over a commutative ring, the notion of classical Krull dimension and the usual prime dimension coincide. This yields that for a multiplication module M, Cl.K.dim(M) exists if and only if M has acc on prime submodules. As an application, we obtain a nice generalization of Cohen's Theorem for multiplication modules. Also, PI-rings whose nonzero modules have zero classical Krull dimension are characterized.