scholarly journals Minimally Generated Modules

1980 ◽  
Vol 23 (1) ◽  
pp. 103-105 ◽  
Author(s):  
W. H. Rant
Keyword(s):  
Local Ring ◽  
Left Module ◽  
Perfect Ring ◽  
Noetherian Module ◽  
Minimal Generator ◽  
Artinian Module ◽  
Maximal Submodule ◽  
Left And Right ◽  
Generator Sets

AbstractA non-zero module M having a minimal generator set contains a maximal submodule. If M is Artinian and all submodules of M have minimal generator sets then M is Noetherian; it follows that every left Artinian module of a left perfect ring is Noetherian. Every right Noetherian module of a left perfect ring is Artinian. It follows that a module over a left and right perfect ring (in particular, commutative) is Artinian if and only if it is Noetherian. We prove that a local ring is left perfect if and only if each left module has a minimal generator set.

Some properties of Artinian modules and applications

2018 ◽  
Vol 17 (02) ◽  
pp. 1850019
Author(s):  
Tran Nguyen An
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Vanishing Theorem ◽  
Strong Connection ◽  
Artinian Modules ◽  
Local Cohomology Modules ◽  
Artinian Module ◽  
Base Ring ◽  
System Of Parameters ◽  
Cohomology Modules

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.

Artinian modules over commutative rings

1992 ◽  
Vol 111 (1) ◽  
pp. 25-33 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Matlis Duality ◽  
Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

Hilbert–Samuel multiplicity and Northcott’s inequality relative to an Artinian module

2019 ◽  
Vol 30 (02) ◽  
pp. 379-396
Author(s):  
V. H. Jorge Pérez ◽  
T. H. Freitas
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Artinian Modules ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Artinian Module ◽  
Samuel Multiplicity

Let [Formula: see text] be a commutative quasi-local ring (with identity [Formula: see text]), and let [Formula: see text] be an [Formula: see text]-ideal such that [Formula: see text]. For [Formula: see text] an Artinian [Formula: see text]-module of N-dimension [Formula: see text], we introduce the notion of Hilbert-coefficients of [Formula: see text] relative to [Formula: see text] and give several properties. When [Formula: see text] is a co-Cohen–Macaulay [Formula: see text]-module, we establish the Northcott’s inequality for Artinian modules. As applications, we show some formulas involving the Hilbert coefficients and we investigate the behavior of these multiplicities when the module is the local cohomology module.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111 ◽  
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

scholarly journals Cotorsion radicals and projective modules

1971 ◽  
Vol 5 (2) ◽  
pp. 241-253 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
Projective Module ◽  
Projective Modules ◽  
Left Module ◽  
Perfect Ring ◽  
Artinian Rings ◽  
Trace Ideal ◽  
Dual Notion ◽  
Rings Of Quotients ◽  
Torsion Radical ◽  
Natural Way

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

scholarly journals On quasi-duo rings

1995 ◽  
Vol 37 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Hua-Ping Yu
Keyword(s):  
Natural Number ◽  
Left Ideal ◽  
Commutative Rings ◽  
Noncommutative Rings ◽  
Perfect Ring ◽  
Maximal Submodule ◽  
Bass Conjecture ◽  
Infinite Sets

Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

scholarly journals Generalized lifting modules

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Yongduo Wang ◽  
Nanqing Ding
Keyword(s):  
Exact Sequence ◽  
Finitely Generated ◽  
Left Module ◽  
Noetherian Module

We introduce the concepts of lifting modules and (quasi-)discrete modules relative to a given left module. We also introduce the notion of SSRS-modules. It is shown that (1) ifMis an amply supplemented module and0→N′→N→N″→0an exact sequence, thenMisN-lifting if and only if it isN′-lifting andN″-lifting; (2) ifMis a Noetherian module, thenMis lifting if and only ifMisR-lifting if and only ifMis an amply supplemented SSRS-module; and (3) letMbe an amply supplemented SSRS-module such thatRad(M)is finitely generated, thenM=K⊕K′, whereKis a radical module andK′is a lifting module.

Reduction of Ideals Relative to an Artinian Module and the Dual of Burch’s Inequality

2019 ◽  
Vol 26 (01) ◽  
pp. 113-122 ◽  
Author(s):  
Fatemeh Cheraghi ◽  
Amir Mafi
Keyword(s):  
Local Ring ◽  
Artinian Module ◽  
Minimal Reduction

Let (A, 𝔪) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dim M = d. If I is an ideal of A with ℓ(0 :M I) < ∞, then we show that for a minimal reduction J of I, (0 :M JI) = (0 :M I2) if and only if [Formula: see text] for all n ≥ 0. Moreover, we study the dual of Burch’s inequality. In particular, the Burch’s inequality becomes an equality if G(I, M) is co-Cohen-Macaulay.

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