scholarly journals Proper classes generated by τ- closed submodules

2019 ◽  
Vol 27 (3) ◽  
pp. 83-95
Author(s):  
Yılmaz Durğun ◽  
Ayşe Çobankaya
Keyword(s):  
Proper Class ◽  
Closed Submodules ◽  
Free Modules ◽  
Torsion Modules ◽  
Exact Sequences ◽  
Torsion Free

AbstractThe main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

Localization and class groups of module categories with exactness defects

1993 ◽  
Vol 113 (2) ◽  
pp. 233-251 ◽  
Author(s):  
D. Holland ◽  
S. M. J. Wilson
Keyword(s):  
Exact Sequence ◽  
Class Group ◽  
Grothendieck Group ◽  
Module Category ◽  
Class Groups ◽  
Definition Of ◽  
Free Modules ◽  
Exact Sequences ◽  
Torsion Free

AbstractWe present a new way of forming a grothendieck group with respect to exact sequences. A ‘defect’ is attached to each non-split sequence and the relation that would normally be derived from a collection of exact sequences is only effective if the (signed) sum of the corresponding defects is zero. The theory of the localization exact sequence and, in particular, of the relative group in this sequence is developed. The (‘locally free’) class group of a module category with exactness defect is defined and an idèlic formula for this is given. The role of torsion and of torsion-free modules is investigated. One aim of the work is to enhance the locally trivial, ‘class group’, invariants obtainable for a module while keeping to a minimum the local obstructions to the definition of such invariants.

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.

When Flats are Torsion Free

1974 ◽  
Vol 17 (4) ◽  
pp. 559-561
Author(s):  
S. Page
Keyword(s):  
Abelian Groups ◽  
Torsion Theory ◽  
Free Modules ◽  
Torsion Modules ◽  
Torsion Free

Given a complete Serre class τ this determines a torsion theory with T the class of torsion modules. It also determines the torsion free modules. For the classical torsion in the category of abelian groups the torsion free modules are flat and visa-versa. Which rings are characterized by this property? More precisely: Which rings admit a torsion theory for which the concepts of torsion free and flat are equivalent? We also dispose of the cases when R admits a toision theory for which torsion free is equivalent to injective and when projective is equivalent to torsion free.

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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