Let A be a finite dimensional algebra. Call an indecomposable module M cool in case $\Omega^{i}(M)$ is nonzero and indecomposable for every $i \geq 1$ and $dim(\Omega^{i}(M))$ is bounded.

Questions:

1. In case every simple module is cool, is the algebra selfinjective?

2.In case every simple module is cool, is every indecomposable non-projective module cool?