37 research outputs found
Syzygy Modules and Injective Cogenerators for Noether Rings
In this paper, we focus on -syzygy modules and the injective cogenerator
determined by the minimal injective resolution of a noether ring. We study the
properties of -syzygy modules and a category which includes
the category consisting of all -syzygy modules and their applications on
Auslander-type rings. Then, we investigate the injective cogenerators
determined by the minimal injective resolution of . We show that is
Gorenstein with finite self-injective dimension at most if and only if \id
R\leq n and \fd \bigoplus_{i=0}^n I_i(R)< \infty. Some known results can be
our corollaries
Torsionfree Dimension of Modules and Self-Injective Dimension of Rings
Let be a left and right Noetherian ring. We introduce the notion of the
torsionfree dimension of finitely generated -modules. For any , we
prove that is a Gorenstein ring with self-injective dimension at most
if and only if every finitely generated left -module and every finitely
generated right -module have torsionfree dimension at most , if and only
if every finitely generated left (or right) -module has Gorenstein dimension
at most . For any , we study the properties of the finitely
generated -modules with \Ext_R^i(M, R)=0 for any .
Then we investigate the relation between these properties and the
self-injective dimension of .Comment: 16 pages. The proof of Lemma 3.8 is modified. It has been accepted
for publication in Osaka Journal of Mathematic
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Let A be a finite dimensional algebra over an algebraic closed field k. In this note, we will show that if T is a separating and splitting tilting A-module, then Ï„-complexities of A and B are equal, where B=EndA(T)
Simple modules over Auslander regular rings
In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of
ExtΛ1(S,Λ) is equal to 1 for any simple Λ-module S with gr S = 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings