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-cofinitely supplemented modules

H. Çalışıcı, A. Pancar (2004)

Czechoslovak Mathematical Journal

Let R be a ring and M a right R -module. M is called -cofinitely supplemented if every submodule N of M with M N finitely generated has a supplement that is a direct summand of M . In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R -module is -cofinitely supplemented. In addition, if M has the summand sum...

A note on generalizations of semisimple modules

Engin Kaynar, Burcu N. Türkmen, Ergül Türkmen (2019)

Commentationes Mathematicae Universitatis Carolinae

A left module M over an arbitrary ring is called an ℛ𝒟 -module (or an ℛ𝒮 -module) if every submodule N of M with Rad ( M ) N is a direct summand of (a supplement in, respectively) M . In this paper, we investigate the various properties of ℛ𝒟 -modules and ℛ𝒮 -modules. We prove that M is an ℛ𝒟 -module if and only if M = Rad ( M ) X , where X is semisimple. We show that a finitely generated ℛ𝒮 -module is semisimple. This gives us the characterization of semisimple rings in terms of ℛ𝒮 -modules. We completely determine the structure of these...

f -derivations on rings and modules

Paul E. Bland (2006)

Commentationes Mathematicae Universitatis Carolinae

If τ is a hereditary torsion theory on 𝐌𝐨𝐝 R and Q τ : 𝐌𝐨𝐝 R 𝐌𝐨𝐝 R is the localization functor, then we show that every f -derivation d : M N has a unique extension to an f τ -derivation d τ : Q τ ( M ) Q τ ( N ) when τ is a differential torsion theory on 𝐌𝐨𝐝 R . Dually, it is shown that if τ is cohereditary and C τ : 𝐌𝐨𝐝 R 𝐌𝐨𝐝 R is the colocalization functor, then every f -derivation d : M N can be lifted uniquely to an f τ -derivation d τ : C τ ( M ) C τ ( N ) .

Hyperdéterminant d’un S L 2 -homomorphisme

Jean Vallès (2008)

Annales mathématiques Blaise Pascal

Etant donnés A 1 , , A s ( s 3 ) des S L 2 ( ) -modules non triviaux de dimensions respectives n 1 + 1 n s + 1 (avec n 1 = n 2 + + n s ) et φ ( A 2 A s , A 1 * ) un S L 2 ( ) -homomorphisme, nous montrons que l’hyperdéterminant de φ est nul sauf si les modules A i sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

On some classes of modules

Gonca Güngöroglu, Harmanci, Abdullah (2000)

Czechoslovak Mathematical Journal

The aim of this paper is to investigate quasi-corational, comonoform, copolyform and α -(co)atomic modules. It is proved that for an ordinal α a right R -module M is α -atomic if and only if it is α -coatomic. And it is also shown that an α -atomic module M is quasi-projective if and only if M is quasi-corationally complete. Some other results are developed.

On μ -singular and μ -extending modules

Yahya Talebi, Ali Reza Moniri Hamzekolaee (2012)

Archivum Mathematicum

Let M be a module and μ be a class of modules in Mod - R which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a μ -essential submodule provided it has a non-zero intersection with any non-zero submodule in μ . We define and investigate μ -singular modules. We also introduce μ -extending and weakly μ -extending modules and mainly study weakly μ -extending modules. We give some characterizations of μ -co-H-rings by weakly μ -extending modules. Let R ...

On τ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

In this paper we introduce the concept of τ -extending modules by τ -rational submodules and study some properties of such modules. It is shown that the set of all τ -rational left ideals of R R is a Gabriel filter. An R -module M is called τ -extending if every submodule of M is τ -rational in a direct summand of M . It is proved that M is τ -extending if and only if M = R e j M E ( R / τ ( R ) ) N , such that N is a τ -extending submodule of M . An example is given to show that the direct sum of τ -extending modules need not be τ -extending....

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