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(1,4)-groups with homocyclic regulator quotient of exponent p³

David M. Arnold, Adolf Mader, Otto Mutzbauer, Ebru Solak (2015)

Colloquium Mathematicae

The class of almost completely decomposable groups with a critical typeset of type (1,4) and a homocyclic regulator quotient of exponent p³ is shown to be of bounded representation type. There are precisely four near-isomorphism classes of indecomposables, all of rank 6.

A categorification of the square root of -1

Yin Tian (2016)

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 representation-finite algebras

Andrzej Skowroński, M. Wenderlich (1991)

Fundamenta Mathematicae

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.

A class of quasitilted rings that are not tilted

Riccardo Colpi, Kent R. Fuller, Enrico Gregorio (2006)

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 classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations

Justyna Kosakowska (2001)

Colloquium Mathematicae

Assume that K is an arbitrary field. Let (I, ⪯) be a two-peak poset of finite prinjective type and let KI be the incidence algebra of I. We study sincere posets I and sincere prinjective modules over KI. The complete set of all sincere two-peak posets of finite prinjective type is given in Theorem 3.1. Moreover, for each such poset I, a complete set of representatives of isomorphism classes of sincere indecomposable prinjective modules over KI is presented in Tables 8.1.

A computation of positive one-peak posets that are Tits-sincere

Marcin Gąsiorek, Daniel Simson (2012)

Colloquium Mathematicae

A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix A ( ) is ℤ-congruent to its transpose A t r is also discussed. An affirmative answer is given for the incidence matrices C I and the Tits matrices C ̂ I of positive one-peak posets I.

A density theorem for algebra representations on the space (s)

W. Żelazko (1998)

Studia Mathematica

We show that an arbitrary irreducible representation T of a real or complex algebra on the F-space (s), or, more generally, on an arbitrary infinite (topological) product of the field of scalars, is totally irreducible, provided its commutant is trivial. This provides an affirmative solution to a problem of Fell and Doran for representations on these spaces.

A duality result for almost split sequences

Lidia Hügel, Helmut Valenta (1999)

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

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

We investigate the category mod Λ of finite length modules over the ring Λ = A k Σ , where Σ 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 E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that mod Λ j b a d h b o x P j . As a consequence we can (1) construct for each elementary k -algebra A over a finite field k a nonartinian noetherian ring Λ such that mod A mod Λ , (2) find twisted...

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