… We propose to call an artin algebra A left weakly Gorenstein provided that … -projective module
is Gorenstein-projective. (And A is said to be right weakly Gorenstein if its opposite algebra …

… on given modules over a graded Artin algebra-we … modules are gradable. It follows immediately
(see Section 3) that simple modules and projective modules over a graded Artin algebra …

… artin algebras A with $$\mathcal{L}_A \cup \mathcal{R}_A$$ co-finite in ind A, quasi-tilted
algebras and generalized double tilted algebras… algebras exhaust the class of all artin algebras …

… modules over arbitrary artin algebras. As would be expected, the preprojective and preinjective
modules … A consisting of all the indecomposable projective modules in mod A. Then it is …

… Proof Let O,, i = 1 ,..., 1 , be the Auslander-Reiten orbits of indecomposable projective modules.
There exist natural numbers nip ..., nl and ml , ..., ml with dim T~D%P = dim P + mih for P …

… of hereditary algebras in terms of the growth number [16]. The main purpose of this paper is
to describe all artin algebras for which every indecomposable projective module is directing. …

… As a first application we see that if A is an artin algebra whose … computing the projective
resolutions of the simple A-modules. … Let A be an artin algebra whose quiver is given by a tree T …

… 23() ARTIN A NIX ZHANG We define the associated projective scheme to be the pair proj A
… and / is the object determined by the right module A , . We also have the autoequivalence s of …

… Applying induction to the 0/aO3-projective module P/FP, we see that the latter is generated
by (n - 1 - r) . r + 1/2 [r(r + 3)] elements, if r ? n - 1, and by 1 + 1/2[(n - 1)(n + 2)] elements if r = n…

… Our method here works for arbitrary Artin rings, rather than Artin algebras, so we show that
each Artin ring … S, we have a chain of the indecomposable projective modules with socle S : …