Let $\Lambda$ be an artin algebra.
- If $M$ is a finitely generated $\Lambda$-module with Loewy length 2 and finite projective dimension. How to get the exact sequence $$0 \rightarrow A \rightarrow P/rad^2P \rightarrow M \rightarrow 0$$ where $P$ is the projective cover of $M$ and $A$ is semisimple?(I just know that $P/radP \cong M/radM$)
- Suppose that $(rad \Lambda)^3=0$, for any $\Lambda$-module $M$, $\Omega M$ is the first syzygy of $M$. Then how to get $\Omega M$ has Loewy length at most 2?