scholarly journals On the Goldie dimension of injective modules

1998 ◽  
Vol 41 (2) ◽  
pp. 265-275 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Pedro A. Guil Asensio
Keyword(s):  
Independent Set ◽  
Uniqueness Condition ◽  
Finitely Generated ◽  
Goldie Dimension ◽  
Injective Modules ◽  
Continuous Module

Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if M satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.

scholarly journals Relative injectivity and CS-modules

1994 ◽  
Vol 17 (4) ◽  
pp. 661-666
Author(s):  
Mahmoud Ahmed Kamal
Keyword(s):  
Direct Decomposition ◽  
Injective Hull ◽  
Extra Condition ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules ◽  
Semisimple Module ◽  
Continuous Module

In this paper we show that a direct decomposition of modulesM⊕N, withNhomologically independent to the injective hull ofM, is a CS-module if and only ifNis injective relative toMand both ofMandNare CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

scholarly journals Injective Modules Over Non-Artinian Serial Rings

1988 ◽  
Vol 44 (2) ◽  
pp. 242-251 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Affirmative Answer ◽  
Primitive Idempotent ◽  
Injective Hull ◽  
Finitely Generated ◽  
Serial Ring ◽  
Direct Sum ◽  
Injective Modules

AbstractA module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

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.

scholarly journals RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

2012 ◽  
Vol 54 (3) ◽  
pp. 605-617 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
NOYAN ER ◽  
NİL ORHAN ERTAŞ
Keyword(s):  
Prime Ideals ◽  
Cyclic Module ◽  
Direct Sums ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Injective Modules ◽  
Dedekind Domains ◽  
Singular Ring

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

scholarly journals Goldie dimensions of quotient modules

2001 ◽  
Vol 71 (1) ◽  
pp. 11-19
Author(s):  
John Dauns
Keyword(s):  
Associative Ring ◽  
Infinite Cardinal ◽  
Goldie Dimension ◽  
Injective Modules ◽  
Subdirect Products ◽  
Quotient Modules ◽  
Quotient Property

AbstractFor an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

On rings with cyclic almost-injective modules

2021 ◽  
pp. 2250180
Author(s):  
Adel Nailevich Abyzov ◽  
Truong Cong Quynh
Keyword(s):  
Finitely Generated ◽  
Injective Modules ◽  
Finite Set

It is shown that every finitely generated right [Formula: see text]-module is almost injective if and only if every cyclic right [Formula: see text]-module is almost injective, if and only if [Formula: see text] is a right [Formula: see text]-ring with [Formula: see text] and there is a finite set of orthogonal idempotents [Formula: see text] in [Formula: see text] such that [Formula: see text] is an injective local right [Formula: see text]-module of length two for every [Formula: see text] and [Formula: see text].

Modules which are invariant under nilpotents of their envelopes and covers

2020 ◽  
pp. 2150218
Author(s):  
Truong Cong Quynh ◽  
Adel Abyzov ◽  
Dinh Duc Tai
Keyword(s):  
Injective Envelope ◽  
Finitely Generated ◽  
Injective Modules ◽  
Semiprimary Ring ◽  
Nilpotent Endomorphism

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with [Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

An alternative perspective on flatness of modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650145 ◽  
Author(s):  
Yılmaz Durğun
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Injective Modules ◽  
Alternative Perspective ◽  
Necessary And Sufficient

Given modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be absolutely [Formula: see text]-pure if [Formula: see text] is a monomorphism for every extension [Formula: see text] of [Formula: see text]. For a module [Formula: see text], the absolutely pure domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is absolutely [Formula: see text]-pure. As an opposite to flatness, a module [Formula: see text] is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and off-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.

scholarly journals KAJIAN KEINJEKTIFAN MODUL (MODUL INJEKTIF, MODUL INJEKTIF LEMAH, MODUL MININJEKTIF)

2014 ◽  
Vol 9 (1) ◽  
Author(s):  
Baidowi Baidowi1 ◽  
Yunita Septriana Anwar
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Injective Modules ◽  
Simple Ideal

Abstrak. Diberikan  adalah -modul. Modul  dikatakan injektif jika untuk setiap monomorfisma   dan setiap homomorfisma  terdapat homomorfisma   sedemikian hingga . Modul  dikatakan injektif-lemah jika  adalah  -injektif lemah untuk setiap modul    yang dibangun berhingga. Sedangkan  dikatakan mininjektif jika untuk setiap homomorfisma dari  dengan  ideal sederhana dari , terdapat homomorfisma  sedemikian hingga . Kajian keinjektifan dalam tulisan ini meliputi modul injektif, modul injektif-lemah, dan modul mininjektif yang mengkaji karakterisasi dari masing-masing modul. Khusunya ketiganya memiliki karakterisasi yang khusus pada jumlahan tak berhingganya.Kata Kunci : Modul injektif, modul injektif-lemah, modul mininjektifAbstract. Let  be an -module. An -module  is called injective if for any monomrphism  and for any homomorphism  there exists a homomorphism  such that . We say that an -module  is weakly-injective if  is weakly -injective for every finitely generated module . An -module  is called mininjective if every homomorphism , there exists a homomorphism  such that , with  is simple ideal of . In this paper, we give some characterizations and properties of injective modules, weakly-injective modules, and mininjective modules. In particular, they have different characterizations for their infinite direct sum.Keywords : Injective modules, weakly-injective modules, mininjective modules

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