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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...

### $\bigcap$-compact modules

Commentationes Mathematicae Universitatis Carolinae

The duals of $\cup$-compact modules are briefly discussed.

### $\left(n,d\right)$-injective covers, $n$-coherent rings, and $\left(n,d\right)$-rings

Czechoslovak Mathematical Journal

It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $\left(n,d\right)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $\left(n,0\right)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $\left(n,0\right)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $\left(n,d\right)$-rings, von Neumann regular rings and semisimple rings....

### A bicategorical approach to static modules.

Theory and Applications of Categories [electronic only]

### A brief review of abelian categorifications.

Theory and Applications of Categories [electronic only]

### A canonical directly infinite ring

Czechoslovak Mathematical Journal

Let $ℕ$ be the set of nonnegative integers and $ℤ$ the ring of integers. Let $ℬ$ be the ring of $N×N$ matrices over $ℤ$ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $ℬ$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $ℱ$ consisting of...

### A categorification of the square root of -1

Fundamenta Mathematicae

We give a graphical calculus for a monoidal DG category ℐ whose Grothendieck group is isomorphic to the ring ℤ[√(-1)]. We construct a categorical action of ℐ which lifts the action of ℤ[√(-1)] on ℤ².

### A characterization of discrete linearly compact rings by means of a duality

Rendiconti del Seminario Matematico della Università di Padova

### A Characterization Of Primal Noetherian Rings.

Revista colombiana de matematicas

### A characterization of the Artinian modules.

Divulgaciones Matemáticas

### A Class of Balanced Non-Uniserial Rings.

Mathematische Annalen

### A class of quasitilted rings that are not tilted

Colloquium Mathematicae

Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

### A class of rings which are algebraic over the integers.

International Journal of Mathematics and Mathematical Sciences

### A classification of symmetric algebras of strictly canonical type

Colloquium Mathematicae

In continuation of our article in Colloq. Math. 116.1, we give a complete description of the symmetric algebras of strictly canonical type by quivers and relations, using Brauer quivers.

### A decomposition theorem for comodules

Compositio Mathematica

### A duality result for almost split sequences

Colloquium Mathematicae

Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.

### A family of noetherian rings with their finite length modules under control

Czechoslovak Mathematical Journal

We investigate the category $\text{mod}\Lambda$ of finite length modules over the ring $\Lambda =A{\otimes }_{k}\Sigma$, where $\Sigma$ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module ${E}_{j}$ gives rise to a quasiprogenerator ${P}_{j}=A\otimes {E}_{j}$. By a result of K. Fuller, ${P}_{j}$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq {\coprod }_{j}badhbox{P}_{j}$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda$ such that $\text{mod}A\simeq \text{mod}\Lambda$, (2) find twisted...

### A functional equation for quadratic forms.

Aequationes mathematicae

Semigroup forum

### A functor between categories of regular modules for wild hereditary algebras.

Mathematische Annalen

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