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We show that a right artinian ring R is right self-injective if and only if ψ(M ) = 0 (or equivalently φ(M ) = 0) for all finitely generated right R-modules M , where ψ, φ : mod R → N are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if φ(M ) = 0 for all finitely generated right Λ-modules M .

Let R be a finite dimensional k-algebra over an algebraically closed field k and mod R be the category of all finitely generated left R-modules. For a given full subcategory X of mod R, we denote by pfd X the projective finitistic dimension of X . That is, pfd X := sup{pd X: X ∈ X and pd X < ∞}.It was conjectured by H. Bass in the 60's that the projective finitistic dimension pfd(R) := pfd(mod R) has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201-204] a function Ψ : mod R → N, which turned out to be useful to prove that pfd(R) is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of mod R instead of a class of algebras. That is, we suggest to take the class of categories F (θ), of θ-filtered R-modules, for all stratifying systems (θ, ) in mod R. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension.

We show that an Artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with a vanishing radical cube.

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