Σ-pure-injective Modules and Artinian Modules

Module Theory ◽  
1998 ◽  
pp. 237-266
Author(s):  
Alberto Facchini
Keyword(s):  
Artinian Modules ◽  
Injective Modules

Decomposition of totally transcendental modules

1980 ◽  
Vol 45 (1) ◽  
pp. 155-164 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Artinian Modules ◽  
Injective Modules ◽  
Unique Decomposition ◽  
Special Cases ◽  
Decomposition Theorems ◽  
Coherent Rings

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.

On the flat length of injective modules

2011 ◽  
Vol 84 (2) ◽  
pp. 408-432 ◽  
Author(s):  
Ioannis Emmanouil ◽  
Olympia Talelli
Keyword(s):  
Injective Modules

On GI-Injective Modules

2012 ◽  
Vol 40 (10) ◽  
pp. 3841-3858 ◽  
Author(s):  
Zenghui Gao
Keyword(s):  
Injective Modules

scholarly journals Injective modules under faithfully flat ring extensions

2015 ◽  
Vol 144 (3) ◽  
pp. 1015-1020 ◽  
Author(s):  
Lars Winther Christensen ◽  
Fatih Köksal
Keyword(s):  
Injective Modules ◽  
Flat Ring ◽  
Ring Extensions

Ramification Indices and Injective Modules

1975 ◽  
Vol s2-11 (3) ◽  
pp. 267-275 ◽  
Author(s):  
Rodney Y. Sharp
Keyword(s):  
Injective Modules

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 Injective modules for group rings and Gorenstein orders

1973 ◽  
Vol 24 (3) ◽  
pp. 465-472 ◽  
Author(s):  
K.W Roggenkamp
Keyword(s):  
Group Rings ◽  
Injective Modules

Injective modules and prime ideals

1981 ◽  
Vol 9 (9) ◽  
pp. 989-999 ◽  
Author(s):  
P.F. Smith
Keyword(s):  
Prime Ideals ◽  
Injective Modules

Generalized Co-Cohen–Macaulay and Co-Buchsbaum Modules

2007 ◽  
Vol 14 (02) ◽  
pp. 265-278
Author(s):  
Nguyen Tu Cuong ◽  
Nguyen Thi Dung ◽  
Le Thanh Nhan
Keyword(s):  
Artinian Modules ◽  
Basic Properties ◽  
Local Homology

We study two classes of Artinian modules called co-Buchsbaum modules and generalized co-Cohen–Macaulay modules. Some basic properties and characterizations of these modules in terms of 𝔮-weak co-sequences, co-standard sequences, multiplicity, local homology modules are presented.

Sign in / Sign up

Export Citation Format

Share Document