Idempotent ideals and the Igusa-Todorov functions

Let $\mathrm{\Lambda }$ be an artin algebra and $\mathfrak{A}$ a two-sided idempotent ideal of $\mathrm{\Lambda }$, that is, $\mathfrak{A}$ is the trace of a projective $\mathrm{\Lambda }$-module $P$ in $\mathrm{\Lambda }$. We consider the categories of finitely generated modules over the associated rings $\mathrm{\Lambda }\mathrm{/}\mathfrak{A},\mathrm{\Lambda }$ and $\mathrm{\Gamma }={\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{\Lambda }}(P{)}^{op}$ and study the relationship between their homological properties via the Igusa-Todorov functions.