Relative exact covers

Ladislav Bican; Blas Torrecillas

Displaying similar documents to “Relative exact covers”

Relatively exact modules

Ladislav Bican (2003)

Commentationes Mathematicae Universitatis Carolinae

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Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice $ℰ_{τ} (M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M / N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M / N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As...

Precovers and Goldie’s torsion theory

Ladislav Bican (2003)

Mathematica Bohemica

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Recently, Rim and Teply , using the notion of $τ$ -exact modules, found a necessary condition for the existence of $τ$ -torsionfree covers with respect to a given hereditary torsion theory $τ$ for the category $R$ -mod of all unitary left $R$ -modules over an associative ring $R$ with identity. Some relations between $τ$ -torsionfree and $τ$ -exact covers have been investigated in . The purpose of this note is to show that if $σ = (𝒯_{σ}, ℱ_{σ})$ is Goldie’s torsion theory and $ℱ_{σ}$ is a precover class, then $ℱ_{τ}$ is a precover class...

Extending modules relative to a torsion theory

Semra Doğruöz (2008)

Czechoslovak Mathematical Journal

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An $R$ -module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $τ$ -extending module, where $τ$ is a hereditary torsion theory on $Mod$ - $R$ . An $R$ -module $M$ is called type 2 $τ$ -extending if every type 2 $τ$ -closed submodule of $M$ is a direct summand of $M$ . If $τ_{I}$ is the torsion theory on $Mod$ - $R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 $τ_{I}$ -extending $R$ -module, then the question of whether...

Precovers

Ladislav Bican, Blas Torrecillas (2003)

Czechoslovak Mathematical Journal

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Let $𝒢$ be an abstract class (closed under isomorpic copies) of left $R$ -modules. In the first part of the paper some sufficient conditions under which $𝒢$ is a precover class are given. The next section studies the $𝒢$ -precovers which are $𝒢$ -covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$ -modules. Especially, several sufficient conditions for the existence of $σ$ -torsionfree and $σ$ -torsionfree $σ$ -injective covers are presented. ...