ON MODULES WHICH ARE SUBISOMORPHIC TO THEIR PURE-INJECTIVE ENVELOPES

2002 ◽  
Vol 01 (03) ◽  
pp. 289-294
Author(s):  
MAHER ZAYED ◽  
AHMED A. ABDEL-AZIZ
Keyword(s):  
Direct Summand ◽  
Global Dimension ◽  
The Other ◽  
Direct Products ◽  
Direct Sums ◽  
Elementary Equivalence ◽  
Injective Modules ◽  
Reduced Products ◽  
Coherent Ring ◽  
Pure Injective Module

In the present paper, modules which are subisomorphic (in the sense of Goldie) to their pure-injective envelopes are studied. These modules will be called almost pure-injective modules. It is shown that every module is isomorphic to a direct summand of an almost pure-injective module. We prove that these modules are ker-injective (in the sense of Birkenmeier) over pure-embeddings. For a coherent ring R, the class of almost pure-injective modules coincides with the class of ker-injective modules if and only if R is regular. Generally, the class of almost pure-injective modules is neither closed under direct sums nor under elementary equivalence. On the other hand, it is closed under direct products and if the ring has pure global dimension less than or equal to one, it is closed under reduced products. Finally, pure-semisimple rings are characterized, in terms of almost pure-injective modules.

Strongly Gorenstein categories

2021 ◽  
pp. 2250213
Author(s):  
Wan Wu ◽  
Zenghui Gao
Keyword(s):  
Direct Summand ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Abelian Category ◽  
Direct Products ◽  
Direct Sums ◽  
Gorenstein Categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

A direct proof of the Feferman-Vaught theorem and other preservation theorems in products

1991 ◽  
Vol 56 (2) ◽  
pp. 632-636 ◽  
Author(s):  
Yiannis Vourtsanis
Keyword(s):  
Direct Proof ◽  
The Other ◽  
Direct Products ◽  
First Order ◽  
New Methods ◽  
Preservation Theorems ◽  
Order Language ◽  
Reduced Products ◽  
The One ◽  
By Products

Here we give short and direct proofs of the Feferman-Vaught theorem and other preservation theorems in products of structures. In 1952, Mostowski [5] first showed the preservation of ≡ωω by direct powers of structures. Subsequently, in 1959, Feferman and Vaught [2] proved the preservation of ≡ωω by arbitrary direct products and also by reduced products with respect to cofinite filters. In 1962, Frayne, Morel and Scott [3] noticed that the results extend to arbitrary reduced products. In 1970, Barwise and Eklof [1] showed the preservation of ≡∞λ by products and in 1971 Malitz [4] showed the preservation of ≡κλ with κ strongly inaccessible (or ∞) by products. Below, we give short proofs of the above results. The ideas used here have initiated, in [7], [8], [9], [10], [11], the introduction of several new methods in the theory of products, which on the one hand give new, direct proofs of the known results in the area, including generalizations or strengthenings of some of those, and, on the other hand, give several new results as well in the theory of products and related areas.Below, L denotes a (first order) language, and by a structure we mean an L-structure. 0 and 1 denote the logically valid and false sentence, respectively. We may write ā ∈ A for ā ∈ An for some n. Also, the values 1 and 2 of a parameter l in the definitions below express a duality corresponding to disjunctive and conjunctive forms.

scholarly journals Direct sums of indecomposable injective modules

2000 ◽  
Vol 62 (1) ◽  
pp. 57-66
Author(s):  
Sang Cheol Lee ◽  
Dong Soo Lee
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules

This paper proves that every direct summand N of a direct sum of indecomposable injective submodules of a module is the sum of a direct sum of indecomposable injective submodules and a sum of indecomposable injective submodules of Z2(N).

Four concepts from “geometrical” stability theory in modules

1992 ◽  
Vol 57 (2) ◽  
pp. 724-740 ◽  
Author(s):  
T. G. Kucera ◽  
M. Prest
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
The Other ◽  
General Stability ◽  
Injective Modules ◽  
Algebraic Problem

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective 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 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 Essentially closed radical classes

1983 ◽  
Vol 35 (1) ◽  
pp. 132-142 ◽  
Author(s):  
N. V. Loi
Keyword(s):  
The Other ◽  
Radical Class ◽  
Direct Sums ◽  
Ring Of Integers ◽  
Essential Extensions

AbstractThe main goal of this paper is to describe radical classes closed under essential extensions. It turns out that such classes are precisely the homomorphically closed semisimple classes, and hence a radical class is essentially closed if and only if it is subdirectly closed. Moreover, a class is closed under homomorphic images, direct sums and essential extensions if and only if it is an essentially closed radical class. Also radical classes are investigated which are closed under Dorroh essentially extensions only, such a radical class R consists of idempotent rings provided that R does not contain the ring of integers, meanwhile all the other radicals satisfy this requirement. A description of (hereditary and) Dorroh essentially closed radicals is given in Theorem 4.

On n-Coherent Rings and (n,d)-Injective Modules

2015 ◽  
Vol 22 (02) ◽  
pp. 349-360
Author(s):  
Dongdong Zhang ◽  
Baiyu Ouyang
Keyword(s):  
Injective Modules ◽  
Coherent Ring ◽  
Derived Functors ◽  
Coherent Rings

Let R be a ring, n, d be fixed non-negative integers, [Formula: see text] the class of (n,d)-injective left R-modules, and [Formula: see text] the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then [Formula: see text] if and only if [Formula: see text], if and only if Ext m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if Ext m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on [Formula: see text] by [Formula: see text], and investigate the global right [Formula: see text]-dimension of [Formula: see text] and the global right [Formula: see text]-dimension of [Formula: see text] by right derived functors of − ⊗ −. Some known results are obtained as corollaries.

Sign in / Sign up

Export Citation Format

Share Document