Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras...

In this paper, we prove that unit ideal-stable range condition is right and left symmetric.

A ring Λ satisfies the Generalized Auslander-Reiten Condition ( ) if for each Λ-module M with $Ex{t}^i(M,M\oplus \Lambda )=0$ for all i > n the projective dimension of M is at most n. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type.

The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0\to A\hookrightarrow L\stackrel{\pi }{\to }B\to 0$ of 3-Lie algebras to obtain obstruction classes in $H^1(B,A)$ for a pair of automorphisms in $\mathrm{Aut}\left(A\right)\times \mathrm{Aut}\left(B\right)$ to be inducible from an automorphism of $L$. Application to free nilpotent 3-Lie algebras is discussed.

A class of finite-dimensional algebras whose Auslander-Reiten quivers have starting but not generalized standard components is investigated. For these components the slices whose slice modules are tilting are considered. Moreover, the endomorphism algebras of tilting slice modules are characterized.

Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M{<}_{deg}N$ such that ${}^{t}M{/}^{t+1}M{\simeq }^{t}N{/}^{t+1}N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $di{m}_{k}Hom(M,M)=di{m}_{k}Hom(P,M)$ if and only if M has no proper degeneration $M{<}_{deg}N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results....

Let $S$ and $R$ be two associative rings, let ${}_S{C}_R$ be a semidualizing $(S,R)$-bimodule. We introduce and investigate properties of the totally reflexive module with respect to ${}_S{C}_R$ and we give a characterization of the class of the totally $C_R$-reflexive modules over any ring $R$. Moreover, we show that the totally $C_R$-reflexive module with finite projective dimension is exactly the finitely generated projective right $R$-module. We then study the relations between the class of totally reflexive modules and the Bass class...

The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp [Sh1]. One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes....

We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if $A$, $B$ are flat algebras over a commutative...