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### $\oplus$-cofinitely supplemented modules

Czechoslovak Mathematical Journal

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

### A note on generalizations of semisimple modules

Commentationes Mathematicae Universitatis Carolinae

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

### Étude des sous-modules compléments dans un A-module

Séminaire Dubreil. Algèbre et théorie des nombres

### $f$-derivations on rings and modules

Commentationes Mathematicae Universitatis Carolinae

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

### Hyperdéterminant d’un $S{L}_{2}$-homomorphisme

Annales mathématiques Blaise Pascal

Etant donnés ${A}_{1},\cdots ,{A}_{s}$ ($s\ge 3$) des $S{L}_{2}\left(ℂ\right)$-modules non triviaux de dimensions respectives ${n}_{1}+1\ge \cdots \ge {n}_{s}+1$ (avec ${n}_{1}={n}_{2}+\cdots +{n}_{s}$) et $\phi \in ℒ\left({A}_{2}\otimes \cdots \otimes {A}_{s},{A}_{1}^{*}\right)$ un $S{L}_{2}\left(ℂ\right)$-homomorphisme, nous montrons que l’hyperdéterminant de $\phi$ est nul sauf si les modules ${A}_{i}$ sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

### On relative homotopy groups of modules.

International Journal of Mathematics and Mathematical Sciences

### On some classes of modules

Czechoslovak Mathematical Journal

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

### On the finiteness of the semigroup of conjugacy classes of left ideals for algebras with radical square zero

Colloquium Mathematicae

Let A be a finite-dimensional algebra over an algebraically closed field with radical square zero, and such that all simple A-modules have dimension at most two. We give a characterization of those A that have finitely many conjugacy classes of left ideals.

### On the lattice of left annihilators of certain rings

Fundamenta Mathematicae

### On $\mu$-singular and $\mu$-extending modules

Archivum Mathematicum

Let $M$ be a module and $\mu$ 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 $\mu$-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu$. We define and investigate $\mu$-singular modules. We also introduce $\mu$-extending and weakly $\mu$-extending modules and mainly study weakly $\mu$-extending modules. We give some characterizations of $\mu$-co-H-rings by weakly $\mu$-extending modules. Let $R$...

### On $\tau$-extending modules

Commentationes Mathematicae Universitatis Carolinae

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

### Об одной алгебре на множестве графов.

Sibirskij matematiceskij zurnal

### Примемение теории дивизоров к численному интегрированию периодических функций многих переменных

Matematiceskij sbornik

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