scholarly journals Some examples of modules over Noetherian rings

1982 ◽  
Vol 23 (1) ◽  
pp. 9-13 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Noetherian Ring ◽  
Essential Extension ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Finitely Generated

The purpose of this note is to prove the following result.Theorem 1. Let n be an integer greater than zero. There exists a prime Noetherian ring R of Krull dimension n + 1 and a finitely generated essential extension W of a simple R-module V suchthat(i) W has Krull dimension n, and(ii) W/V is n-critical and cannot be embedded in any of its proper submodules.We refer the reader to [6] for the definition and properties of Krull dimension.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

On Weakly Laskerian and Weakly Cofinite Modules

2012 ◽  
Vol 19 (04) ◽  
pp. 693-698
Author(s):  
Kazem Khashyarmanesh ◽  
M. Tamer Koşan ◽  
Serap Şahinkaya
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Cofinite Modules ◽  
Gorenstein Local Ring

Let R be a commutative Noetherian ring with non-zero identity, 𝔞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module [Formula: see text] is weakly Laskerian for all i < r. Then we prove that [Formula: see text] is also weakly Laskerian and so [Formula: see text] is finite. Moreover, we show that if s is a non-negative integer such that [Formula: see text] is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then [Formula: see text] is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of [Formula: see text] is weakly Laskerian?

Krull Dimension of Injective Modules Over Commutative Noetherian Rings

2005 ◽  
Vol 48 (2) ◽  
pp. 275-282
Author(s):  
Patrick F. Smith
Keyword(s):  
Noetherian Ring ◽  
Integral Domain ◽  
Krull Dimension ◽  
Injective Module ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Injective Modules

AbstractLet R be a commutative Noetherian integral domain with field of fractions Q. Generalizing a forty-year-old theorem of E. Matlis, we prove that the R-module Q/R (or Q) has Krull dimension if and only if R is semilocal and one-dimensional. Moreover, if X is an injective module over a commutative Noetherian ring such that X has Krull dimension, then the Krull dimension of X is at most 1.

Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings

1981 ◽  
Vol 33 (2) ◽  
pp. 325-346 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Additional Assumption ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Local Rings ◽  
Minimal Prime

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.

scholarly journals Bi-artinian noetherian rings

2001 ◽  
Vol 43 (1) ◽  
pp. 9-21
Author(s):  
E. A. Whelan
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Ideal ◽  
Chain Condition ◽  
Noetherian Rings ◽  
Composition Series ◽  
Finitely Generated ◽  
Analogous Statement ◽  
Descending Chain Condition ◽  
Unique Maximal Ideal

A noetherian ring R satisfies the descending chain condition on two-sided ideals (“is bi-artinian”) if and only if, for each prime P ∈ spec(R), R/P has a unique minimal ideal (necessarily idempotent and left-right essential in R/P). The analogous statement for merely right noetherian rings is false, although our proof does not use the full noetherian condition on both sides, requiring only that two-sided ideals be finitely generated on both sides and that R/Q be right Goldie for each Q ∈ spec(R). Examples exist, for each n∈ℕ and in all characteristics, of bi-artinian noetherian domains Dn with composition series of length 2n and with a unique maximal ideal of height n. Noetherian rings which satisfy the related E-restricted bi-d.c.c. do not, in general, satisfy the second layer condition (on either side), but do satisfy the Jacobson conjecture.

scholarly journals Krull dimension and invertible ideals in Noetherian rings

1976 ◽  
Vol 20 (2) ◽  
pp. 81-86 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Cartesian Product ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Infinite Sequences ◽  
Krull Dimensions

In this note we consider the question: If R is a right Noetherian ring and I is an invertible ideal of R, how do the Krull dimensions of various modules, factor rings and over-rings of R, connected with I, compare with the Krull dimension of R? This question is prompted by results in (5) and (6). In comparing the Krull dimension of the ring R with that of the ring R/I, the best result would be that the Krull dimension of the ring R is exactly one greater than that of the ring R/I. This result is not true in general; however, we see, in Theorem 2.4, that if the invertible ideal is contained in the Jacobson radical the result holds. In the general case we find it is necessary to introduce an over-ring T of R generated by the inverse I−1 of I. We then see that the Krull dimension of R is the larger of two possibilities: (a) Krull dimension of R/I plus one or (b) Krull dimension of T. In order to prove this result we construct a strictly increasing map from the poset of right ideals of R to the cartesian product of the poset of right ideals of T with a poset of certain infinite sequences of right ideals of R/I.

Hilbert–Samuel functions of well bifiltered modules

2016 ◽  
Vol 09 (02) ◽  
pp. 1650031 ◽  
Author(s):  
H. Dichi ◽  
D. Sangare
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
General Framework ◽  
Polynomial Function ◽  
Krull Dimension ◽  
Hilbert Functions ◽  
Polynomial Functions ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Base Ring

In an earlier paper, we studied the Hilbert quasi-polynomial functions of finitely generated bigraded modules in the general framework when the base ring is bigraded and generated by finitely many homogeneous elements of arbitrary degrees. In this paper, we introduce the concept of [Formula: see text]-good bifiltration [Formula: see text] on a finitely generated [Formula: see text]-module [Formula: see text], where [Formula: see text] and [Formula: see text] are specified noetherian filtrations on the noetherian ring [Formula: see text]. The bigraded modules associated with such bifiltrations are shown to be finitely generated under reasonable hypotheses. Their Hilbert functions are studied. The Hilbert–Samuel function of [Formula: see text] with respect to the [Formula: see text]-good bifiltration [Formula: see text] of [Formula: see text] is one of them. It is proved, among others, that this function is a quasi-polynomial function in two variables and that if [Formula: see text] is a noetherian local ring and if the filtrations [Formula: see text] and [Formula: see text] are primary filtrations, then its degree equals the Krull dimension of [Formula: see text].

scholarly journals Dimension of finite free complexes over commutative Noetherian rings

2021 ◽  
pp. 11-17
Author(s):  
Lars Christensen ◽  
Srikanth Iyengar
Keyword(s):  
Finite Rank ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Bounded Complex ◽  
Free Modules

Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.

Some Rigidity Results for Highest Order Local Cohomology Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 497-504 ◽  
Author(s):  
Mohammad T. Dibaei ◽  
Siamak Yassemi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Commutative Noetherian Ring ◽  
Rigidity Results ◽  
Local Cohomology Modules ◽  
Formal Behaviour ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a finitely generated R-module of finite Krull dimension n. We describe the (finite) sets [Formula: see text] and [Formula: see text] of primes associated and attached to the highest local cohomology module [Formula: see text] in terms of the local formal behaviour of 𝔞.

Finitely Generated Modules and Connectivity

2006 ◽  
Vol 13 (4) ◽  
pp. 599-606
Author(s):  
Habibollah Ansari-Toroghy ◽  
Reza Ovlyaee-Sarmazdeh
Keyword(s):  
Topological Space ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Zariski Topology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Finitely Generated Modules

Abstract Let 𝑅 be a commutative Noetherian ring and let 𝑀 be a finitely generated 𝑅-module. Let 𝑋 = Spec𝑅(𝑀) be the topological space with Zariski topology. Our main goal in this paper is to describe the connectedness dimension of 𝑅 in terms of Krull dimension of some quotient of 𝑀 and prove that 𝑐(Spec𝑅(𝑀)) = 𝑐(Supp(𝑀)).

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