Self-injective Right Artinian Rings and Igusa Todorov Functions

Abstract

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 .

Relative Igusa-Todorov Functions and Relative Homological Dimensions

“…In such IT -contexts, we develop the theory of the relative IT -functions, and correspondingly, the theory of the relative Igusa-Todorov dimensions. One of the results we got is that Frobenius categories can be characterised by using relative IT -functions, generalising some of the results obtained in [25,28] for quasi-co-Frobenius coalgebras and selfinjective algebras, respectively. Moreover, some important homological dimensions (as the representation dimension introduced by M. Auslander in [3]) can be seen as a particular case of a relative Igusa-Todorov dimension.…”

“…The study of IT -functions is already interesting in itself. Surprisingly, as was discovered by F. Huard and M. Lanzilotta in [28], the Igusa-Todorov functions can be used to characterise self-injective algebras. If we want to go further, a natural question arise: is it possible to define IT -functions in a more general contexts as exact categories?…”

Recollements and homological dimensions

“…These Igusa-Todorov functions determine new homological measures, generalising the notion of projective dimension, and have become a powerful tool in the understanding of the finitistic dimension conjecture [14,22,25]. From [12,13], the φ-dimension of an algebra A is…”

“…The φ-dimension of an algebra A has a strong connection with its global dimension and finitistic dimension: fin.dim(A) ≤ φdim(A) ≤ gl.dim(A) and they all coincide in the case of gl.dim(A) < ∞. Moreover, the φ-dimension can be used to describe selfinjective algebras: an algebra A is selfinjective if and only if φdim(A) = 0 [12]. Recently, various works were dedicated to study and generalise the properties of Igusa-Todorov function and the φdimension [7,12,13,18,25].…”

“…Moreover, the φ-dimension can be used to describe selfinjective algebras: an algebra A is selfinjective if and only if φdim(A) = 0 [12]. Recently, various works were dedicated to study and generalise the properties of Igusa-Todorov function and the φdimension [7,12,13,18,25]. In particular, the φ-dimension of an algebra A was characterised in terms of the bi-functors Ext i A (−, −) and Tor A i (−, −) in [7].…”