The Extension Property in the Category of Direct Sum of Cyclic Torsion-Free Modules over a BFD

2020 ◽  
pp. 313-323
Author(s):  
Seddik Abdelalim ◽  
Abdelhak Chaichaa ◽  
Mostafa El garn
Keyword(s):  
Extension Property ◽  
Direct Sum ◽  
Cyclic Torsion ◽  
Free Modules ◽  
Torsion Free

Injective Hulls of Torsion Free Modules

1971 ◽  
Vol 23 (6) ◽  
pp. 1094-1101 ◽  
Author(s):  
J. Zelmanowitz
Keyword(s):  
Nonzero Element ◽  
Injective Hull ◽  
Direct Products ◽  
Finitely Generated ◽  
Basic Theorem ◽  
Direct Sum ◽  
Free Modules ◽  
Injective Hulls ◽  
First Time ◽  
Torsion Free

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).

Weakly-injective modules over hereditary noetherian prime rings

1995 ◽  
Vol 58 (3) ◽  
pp. 287-297 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth
Keyword(s):  
Simple Module ◽  
Injective Module ◽  
Injective Hull ◽  
Prime Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Hereditary Noetherian Prime Ring ◽  
Free Modules ◽  
Torsion Modules ◽  
Torsion Free

AbstractA module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Torsion and Torsion free Modules

2017 ◽  
pp. 219-233
Author(s):  
Jonathan S. Golan ◽  
Tom Head
Keyword(s):  
Free Modules ◽  
Torsion Free

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

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