共查询到20条相似文献,搜索用时 31 毫秒
1.
An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal. 相似文献
2.
Duong Quôc Viê.t 《Journal of Pure and Applied Algebra》2006,205(3):498-509
Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals. 相似文献
3.
Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by heightN I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included.
Received: 3 July 2005 相似文献
4.
An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/It)=Ass(R/I) for all t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that It has embedded primes is bigger than β1, where β1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If, in addition, I fails to have the packing property, then embedded primes of It do occur when t=β1+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols. 相似文献
5.
Bruce Olberding 《Journal of Pure and Applied Algebra》2008,212(7):1797-1821
Let D be a Noetherian domain of Krull dimension 2, and let H⊆R be integrally closed overrings of D. We examine when H can be represented in the form H=(?V∈ΣV)∩R, with Σ a Noetherian subspace of the Zariski-Riemann space of the quotient field of D. We characterize also the special case in which Σ can be chosen to be a finite character collection of valuation overrings of D. 相似文献
6.
Tony J. Puthenpurakal 《Journal of Pure and Applied Algebra》2007,208(1):159-176
In this paper we introduce a new technique to study associated graded modules. Let (A,m) be a Noetherian local ring with . Our techniques give a necessary and sufficient condition for for all n?0. Other applications are also included; most notable is an upper bound regarding the Ratliff-Rush filtration. 相似文献
7.
We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R-module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras. 相似文献
8.
Hamid Reza Maimani 《Journal of Pure and Applied Algebra》2008,212(1):168-174
Let R be a commutative ring with identity and let I be an ideal of R. Let R?I be the subring of R×R consisting of the elements (r,r+i) for r∈R and i∈I. We study the diameter and girth of the zero-divisor graph of the ring R?I. 相似文献
9.
A. Mimouni 《Journal of Pure and Applied Algebra》2003,177(1):79-93
In this paper we are mainly concerned with TW-domains, i.e., domains in which the w- and t-operations coincide. Precisely, we investigate possible connections with related well-known classes. We characterize the TW-property in terms of divisoriality for Mori domains and Noetherian domains. Specifically, we prove that a Mori domain R is a TW-domain if and only if RM is a divisorial domain for each t-maximal ideal M of R. It turns out that a Mori domain which is a TW-domain is a Strong Mori domain. The last section examines the transfer of the “TW-domain” and “Strong Mori” properties to pullbacks, in order to provide some original examples. 相似文献
10.
Chris Woodcock 《Journal of Pure and Applied Algebra》2007,210(1):193-199
Throughout, all rings R will be commutative with identity element. In this paper we introduce, for each finite group G, a commutative graded Z-algebra RG. This classifies the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element.In the case when G is an elementary Abelian p-group it turns out that RG is closely related to the symmetric algebra over Fp of the dual of G. We intend in subsequent papers to explore the close relationship between G and RG in the case of a general (possibly non-Abelian) group G.Here we show that the Krull dimension of RG is the maximal rank r of an elementary Abelian subgroup E of G unless either E is cyclic or for some such E its normalizer in G contains a non-trivial cyclic group which acts faithfully on E via “scalar multiplication” in which case it is r+1. 相似文献
11.
Babak Jabarnejad 《Journal of Pure and Applied Algebra》2018,222(7):1906-1910
In this paper we describe the defining equations of the multi-Rees algebra over a Noetherian ring R when I is an ideal of linear type. This generalizes a result of Ribbe and recent work of Lin–Polini and Sosa. 相似文献
12.
Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components En, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules En, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread. 相似文献
13.
Christian U. Jensen Søren Jøndrup Anders Thorup 《Journal of Pure and Applied Algebra》2019,223(5):1869-1896
Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras vanishing on . 相似文献
14.
《Quaestiones Mathematicae》2013,36(5):613-629
AbstractLet R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by ΓI (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ? V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of ΓI (R) and the comaximal graph CG2(R) \ J (R) (or CGJ (R) for short) where CG2(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R. 相似文献
15.
Wolfgang Rump 《Algebras and Representation Theory》2006,9(3):227-239
We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal.
Presented by H. Tachikawa 相似文献
16.
Thomas Marley 《manuscripta mathematica》2001,104(4):519-525
Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H
i
I
(M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H
d
− 1
I
(M) has finite support for arbitrary R, I, and M.
Received: 31 October 2000 / Revised version: 8 January 2001 相似文献
17.
It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F -purity. In this note, we prove that if (R,m) is an equidimensional and S2 local ring that admits a canonical ideal I≅ωR such that R/I is F-pure, then R is F-pure. This greatly generalizes one of the main theorems in [2]. We also provide examples to show that not all Cohen–Macaulay F-pure local rings satisfy the above property. 相似文献
18.
In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom
R
(Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext
i
R
(E,M) = 0 for all injective R-modules E and all i > 0.
Received: 24 August 2006 Revised: 30 October 2006 相似文献
19.
Marek Golasiński 《Topology and its Applications》2011,158(14):1756-1761
The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η1-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers. 相似文献
20.
《Journal of Pure and Applied Algebra》1987,47(2):205-212
Ideals whose adic and symbolic topologies are linearly equivalent are characterized in terms of analytic spread and u-essential prime divisors. Using this characterization, under certain conditions on a Noetherian ring R and an ideal I of R it is shown that the I-adic and the I-symbolic topologies are linearly equivalent iff gr(I,R)red is a domain, and locally unmixed rings are characterized as those rings in which the adic and the symbolic topologies of every ideal of the principal class are linearly equivalent. 相似文献