Self-injective right artinian rings and Igusa Todorov functions

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 $ψ, ϕ: \mod R \to \mathbb N$ 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 $M$.

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