We give necessary and sufficient conditions for a wing of an Auslander-Reiten quiver of a selfinjective algebra to be the wing of the radical of an indecomposable projective module. Moreover, a characterization of indecomposable Nakayama algebras of Loewy length ≥ 3 is obtained.
Let be a module and be a class of modules in which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a -essential submodule provided it has a non-zero intersection with any non-zero submodule in . We define and investigate -singular modules. We also introduce -extending and weakly -extending modules and mainly study weakly -extending modules. We give some characterizations of -co-H-rings by weakly -extending modules. Let ...
In this paper we introduce the concept of -extending modules by -rational submodules and study some properties of such modules. It is shown that the set of all -rational left ideals of is a Gabriel filter. An -module is called -extending if every submodule of is -rational in a direct summand of . It is proved that is -extending if and only if , such that is a -extending submodule of . An example is given to show that the direct sum of -extending modules need not be -extending....
We classify one-directed indecomposable pure injective modules over finite-dimensional string algebras.
In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.
Left selfdistributive rings (i.e., ) which are semidirect sums of boolean rings and rings nilpotent of index at most 3 are studied.
Let be a preprojective algebra of type , and let be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories for an injective -module, and we introduce a mutation operation between complete rigid modules in . This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to .
We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing is the path coalgebra of a profinite bound quiver (Q,), and the category C-Comod...
If M is a simple module over a ring R then, by the Schur's lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.