scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

scholarly journals On τ-completely decomposable modules

2004 ◽  
Vol 70 (1) ◽  
pp. 163-175 ◽  
Author(s):  
Septimiu Crivei
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Torsion Theory ◽  
Positive Answer ◽  
Essential Extension ◽  
Injective Module ◽  
Injective Hull ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Torsion Theories

For a hereditary torsion theory τ, a moduleAis called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ringRthat is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.

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.

t-Rickart and Dual t-Rickart Modules

2015 ◽  
Vol 22 (spec01) ◽  
pp. 849-870 ◽  
Author(s):  
Sh. Asgari ◽  
A. Haghany
Keyword(s):  
Direct Summand ◽  
Torsion Module ◽  
Direct Sum ◽  
Rickart Modules ◽  
Baer Modules ◽  
Equivalent Conditions

We introduce the notion of t-Rickart modules as a generalization of t-Baer modules. Dual t-Rickart modules are also defined. Both of these are generalizations of continuous modules. Every direct summand of a t-Rickart (resp., dual t-Rickart) module inherits the property. Some equivalent conditions to being t-Rickart (resp., dual t-Rickart) are given. In particular, we show that a module M is t-Rickart (resp., dual t-Rickart) if and only if M is a direct sum of a Z2-torsion module and a nonsingular Rickart (resp., dual Rickart) module. It is proved that for a ring R, every R-module is dual t-Rickart if and only if R is right t-semisimple, while every R-module is t-Rickart if and only if R is right Σ-t-extending. Other types of rings are characterized by certain classes of t-Rickart (resp., dual t-Rickart) modules.

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

1992 ◽  
Vol 35 (3) ◽  
pp. 511-518
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

Let E be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0:Eα) of E has a secondary representation, and so we can form the finite set AttA(0:Eα) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets is ultimately constant; in [2], we introduced the integral closure a*(E) of α relative to E, and showed that is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then is ultimately constant, and we obtain information about its ultimate constant value.

D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

2020 ◽  
pp. 1-27
Author(s):  
GABRIELLA D′ESTE ◽  
DERYA KESKİN TÜTÜNCÜ ◽  
RACHID TRIBAK
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Rickart Modules

Abstract A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

Σ-Rickart modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050207
Author(s):  
Gangyong Lee ◽  
Mauricio Medina-Bárcenas
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Hereditary Ring ◽  
Direct Sum ◽  
Factor Module ◽  
Rickart Modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

Idempotent Ideals and Noetherian Polynomial Rings

1982 ◽  
Vol 25 (1) ◽  
pp. 48-53 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Power Series ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Power Series Rings ◽  
Descending Chain Condition ◽  
The Ideal

AbstractIf R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.

The first non-isomorphic local cohomology modules with respect to their ideals

2018 ◽  
Vol 17 (12) ◽  
pp. 1850230
Author(s):  
Ali Fathi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Regular Sequences ◽  
Maximal Ideals ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let [Formula: see text] be ideals of a commutative Noetherian ring [Formula: see text] and [Formula: see text] be a finitely generated [Formula: see text]-module. By using filter regular sequences, we show that the infimum of integers [Formula: see text] such that the local cohomology modules [Formula: see text] and [Formula: see text] are not isomorphic is equal to the infimum of the depths of [Formula: see text]-modules [Formula: see text], where [Formula: see text] runs over all prime ideals of [Formula: see text] containing only one of the ideals [Formula: see text]. In particular, these local cohomology modules are isomorphic for all integers [Formula: see text] if and only if [Formula: see text]. As an application of this result, we prove that for a positive integer [Formula: see text], [Formula: see text] is Artinian for all [Formula: see text] if and only if, it can be represented as a finite direct sum of [Formula: see text] local cohomology modules of [Formula: see text] with respect to some maximal ideals in [Formula: see text] for any [Formula: see text]. These representations are unique when they are minimal with respect to inclusion.

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

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