We prove that the component quiver of a connected self-injective artin algebra A of infinite representation type is fully cyclic, that is, every finite set of components of the Auslander-Reiten quiver of A lies on a common oriented cycle in .
Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.
In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as -modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.
The rational completion of an -module can be characterized as a -injective hull of with respect to a (hereditary) torsion functor depending on . Properties of a torsion functor depending on an -module are studied.
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...
Von Neumann regular rings, hereditary rings, semi-simple Artinian rings, self-injective regular rings are characterized. Rings which are either strongly regular or semi-simple Artinian are considered. Annihilator ideals and -regular rings are studied. Properties of WGP-injectivity are developed.
We introduce and study the concepts of weak -injective and weak -flat modules in terms of super finitely presented modules whose projective dimension is at most , which generalize the -FP-injective and -flat modules. We show that the class of all weak -injective -modules is injectively resolving, whereas that of weak -flat right -modules is projectively resolving and the class of weak -injective (or weak -flat) modules together with its left (or right) orthogonal class forms a hereditary...