Let $G$ be a finite nonabelian group, $\mathbb{Z}G$ its associated integral group ring, and $▵\left(G\right)$ its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups $Q_n\left(G\right)={▵}^n\left(G\right)/{▵}^{n+1}\left(G\right)$ is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $\mathbb{Z}\Delta /G$ where $\Delta$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for ‘most’ groups $G$, the category $𝒯$ is standard,i.e.$k$-linearly...

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $\Lambda$ of a $p$-adic analytic group $G$. For $G$ without any $p$-torsion element we prove that $\Lambda$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $\Lambda$-module. This is classical when $G={\mathbb{Z}}_p^k$ for some integer $k\ge 1$, but was previously unknown in the non-commutative case. Then the category of $\Lambda$-modules...

In this paper we consider completely decomposable torsion-free groups and we determine the subgroups which are ideals in every ring over such groups.

For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End{T}_A$, then T(A) is tame if and only if T(B) is tame.

The aim of this note is to give an affirmative answer to a problem raised in [9] by J. Nehring and A. Skowroński, concerning the number of nonstable ℙ₁(K)-families of quasi-tubes in the Auslander-Reiten quivers of the trivial extensions of tubular algebras over algebraically closed fields K.

We give the characterization of the unit group of ${𝔽}_{q}SL(2,{\mathbb{Z}}_5)$, where ${𝔽}_q$ is a finite field with $q=p^k$ elements for prime $p>5,$ and $SL(2,{\mathbb{Z}}_5)$ denotes the special linear group of $2\times 2$ matrices having determinant $1$ over the cyclic group ${\mathbb{Z}}_5$.

We present some results on the location of zeros of regular polynomials of a quaternionic variable. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets of a regular product established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results extend some classical results from complex to the quaternionic setting as well.

We use modules of finite length to compare various generalizations of the classical tilting and cotilting modules introduced by Brenner and Butler [BrBu].

In this paper, we define Gorenstein injective rings, Gorenstein injective modules and their envelopes. The main topic of this paper is to show that if $D$ is a Gorenstein integral domain and $M$ is a left $D$-module, then the torsion submodule $tGM$ of Gorenstein injective envelope $GM$ of $M$ is also Gorenstein injective. We can also show that if $M$ is a torsion $D$-module of a Gorenstein injective integral domain $D$, then the Gorenstein injective envelope $GM$ of $M$ is torsion.