Lifting idempotent endomorphisms of the quotient of a projective module

Question

Let $E \cong P/M$, where $P$ is a projective module and $M$ is a submodule. Suppose $\beta: E \to E$ is an idempotent endomorphism of $E$. Can we lift $\beta$ to an idempotent endomorphism of $P$?

Using the lifting property of projective modules, I think we can find $\alpha$ making this diagram commute.

However, I'm struggling to prove/disprove that we can lift $\beta$ to another idempotent.

[I'm starting to learn about projective covers and thought of this when trying to get accustomed to working with lifting properties; apologies if this is a really trivial question.]

Answer 1

We cannot lift $\beta$ to an idempotent map in general. Consider the case of $\mathbb{Z}$-modules. Let $P=\mathbb{Z},M=6\mathbb{Z}$. Note that $P$ is free, hence projective. Let $\beta:E\to E$ be multiplication by $\bar{3}$.

This is an idempotent endomorphism of $E$ since $\bar{3}^2=\bar{3}$. Now, the only idempotent endomorphisms of $\mathbb{Z}$ are the zero map and $Id_\mathbb{Z}$ (since endomorphisms of $\mathbb{Z}$ are given by multiplication by some integer $m\in\mathbb{Z}$), none of them inducing $\beta$ on $E$.