scholarly journals Rings characterized by their weakly-injective modules

1992 ◽  
Vol 34 (3) ◽  
pp. 349-353 ◽  
Author(s):  
Sergio R. López-Permouth
Keyword(s):  
Injective Hull ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Arbitrary Ring ◽  
Injective Modules

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the paper, the notation A ⊂' B will mean that A is an essential submodule of the module B. Given an arbitrary ring R and R-modules M and N, we say that M is weakly N-injective if and only if every map φ:N → E(M) from N into the injective hull E(M) of M may be written as a composition σ〫 , where :N→M and σ:M→E(M) is a monomorphism. This is equivalent to saying that for every map φ:N→E(M), there exists a submodule X of E(M), isomorphic to M, such that φ(N) is contained in X. In particular, M is weakly R-injective if and only if, for every x ∈ E(M), there exists X ⊂ E(M) such that x ∈ X ≌ M. We say that M is weakly-injective if and only if it is weakly N-innjective for every finitely generated module N. Clearly, M is weakly-injective if and only if, for every finitely generated submodule N of E(M), there exists X ⊂ E(M) such that N ⊂ X ≌ M.

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 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

scholarly journals On the Sum of Homological Dimension and Codimension of Modules over a Semi-local Ring

1968 ◽  
Vol 33 ◽  
pp. 165-172
Author(s):  
Samuel S.H. Young
Keyword(s):  
Local Ring ◽  
Global Dimension ◽  
Homological Dimension ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Arbitrary Ring ◽  
Local Case

Let M be a finitely generated module over a regular local ring R. It is well known that the sum of homological dimension and codimension of M is equal to the global dimension of R. For modules over an arbitrary ring, this is in general not true. The purpose of this paper is to investigate the properties of such sums in the semi-local case.

On a Class of Automaton Algebras

2016 ◽  
Vol 60 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

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.

Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

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