Self-injective right artinian rings and Igusa Todorov functions

TL;DR

This paper characterizes right self-injective artinian rings using Igusa-Todorov functions, showing that these functions vanish precisely when the ring is self-injective, providing a new criterion for such rings.

Contribution

It establishes a novel characterization of right self-injective artinian rings via Igusa-Todorov functions, linking module-theoretic properties to ring injectivity.

Findings

Right self-injective rings correspond to vanishing Igusa-Todorov functions.

Igusa-Todorov functions provide a new criterion for self-injectivity.

The characterization applies to artinian rings and artin algebras.

Abstract

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 $M$.