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