Dimension of Crystalline Graded Rings (original) (raw)
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Valuation Derived from Graded Ring and Module and Krull Dimension Properties
In this paper we show if R is a graded ring then we can define a valuation on R induced by graded structure, and we prove some properties and relations for R. Later we show that if R is a graded ring and M a graded R-module then there exists a valuation on of M which is derived from graded structure and also we prove some properties and relations for R. In the following we give a new method for finding the Kurll dimension of a valuation ring.
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
The Center of Crystalline Graded Rings
Eprint Arxiv 0903 4640, 2009
In the first section of the paper, we will give some basic definitions and properties about Crystalline Graded Rings. In the following section we will provide a general description of the center. Afterwards, the case where the grading group is Abelian finite will be handled. The center will have some properties of a crystalline graded ring, but not all. We will call this Arithmetically Crystalline Graded. The center is crystalline graded if the part of degree zero is a principal ideal domain. The last section deals with the case where the grading group is non-Abelian finite. Since this situation is much more complicated than the Abelian case, we primarily focus on the conditions to have a trivial center. The fact that the center is Arithmetically Crystalline Graded also holds in this case.
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.
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).
Associated graded rings of one-dimensional
2010
Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).
On the depth of the associated graded ring
Proceedings of the American Mathematical Society, 1995
Let (R, m) be a Cohen-Macaulay local ring of positive dimension d, let I be an m − m - primary ideal of R. In this paper we individuate some conditions on I that allow us to determine a lower bound for depth gr I ( R ) {\\text {gr}_I}(R) . It is proved that if J ⊆ I J \\subseteq I is a minimal reduction of I such that λ ( I 2 ∩ J / I J ) = 2 \\lambda ({I^2} \\cap J/IJ) = 2 and I n ∩ J = I n − 1 J {I^n} \\cap J = {I^{n - 1}}J for all n ≥ 3 n \\geq 3 , then depth gr I ( R ) ≥ d − 2 {\\text {gr}_I}(R) \\geq d - 2 ; let us remark that λ \\lambda denotes the length function.
Associated graded rings of one-dimensional analytically irreducible rings II
Journal of Algebra, 2011
Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).