I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows:

Recently I came across the notion of Endo-Trivial modules (out of brevity's sake e-t), and was a surprise for me that there is a huge (and rather complicated) theory behind them. I recall that an e-t module is a finitely-generated $\mathbb{K}G$-module over a field of positive characteristic $\mathbb{K}$ and say $G$ a $p$-group, such that $M^{*} \otimes M \cong \mathbb{K} \oplus \textit{(proj)}$, where $(proj)$ states for some projective $\mathbb{K}G$-module. Apparently there should be a kind of modular represenation theoretic argument to study those objects, however isn't clear to me. Do you know what's the initiative behind their study?

Also, there is a decomposition of those modules always, namely
$$M = M_0 \oplus \textit{(proj)},$$
for some indecomposable submodule $M_0$, and some projective. However this isn't clear either. The latter should be some kind of version of Krull-Schmidt theorem, since $M$ is a f.g module over an *Artinian Ring* (and therefore of finite length), hence a decomposition into indecomposables exists and is unique (up to isomorphism). However the theorem doesn't mention anything about projectivity for the indecomposables, so I can't come up with a better idea unfortunately.

Could you please help me out?

"...can have at most one non-projective indecomposable summand", because it isn't quite clear to me theat most onepart. And regarding the aim of their study, what you imply is that because are occurring naturally in a certain setup we want to understand through their structure collectively, the representation theory of $G$, right? $\endgroup$ – mayer_vietoris Jun 18 '17 at 13:46