Let $\beta$ a $\Lambda$-homomorphism is nilpotent then $\operatorname{Ext}^k(\beta , M)$ is nilpotent, for all $M$ $\in$ $\operatorname{mod}(\Lambda)$

Question

I want to prove that if a homomorphism of $\Lambda$-modules, $\beta$, is nilpotent then $\operatorname{Ext}^k_\Lambda(\beta , M)$ is nilpotent, for all $M \in \operatorname{mod}(\Lambda)$. Where $\Lambda$ is a $K$-algebra of Artin finite dimensional.

the ideia is to use that $\operatorname{Ext}$ is a additive functor.

Answer 1

Functoriality implies that $\operatorname{Ext}(\beta,M)^i=\operatorname{Ext}(\beta^i,M)$, so if one power of $\beta$ is zero, the same power of $\operatorname{Ext}(\beta,M)$ is zero, because $\operatorname{Ext}(\text{zero map},M)=0$.