We will call artin R-algebra $\Lambda$ basic if $\Lambda \simeq \coprod\limits_{i=1}^nP_i$ where $P_i$ are nonisomorphic indecomposable projective R-modules. Let $\Lambda$ be a basic artin R-algebra, let $\Lambda'$ be an R-subalgebra of $\Lambda$, let $e$ be an idempotent in $\Lambda$.
1) Prove that $\Lambda'$ is basic;
2) Prove that $e\Lambda e$ is basic.
Can you please help me to prove these two propositions?
I know there is a bijection between the set of indecomposable projective modules and the set of simple ones. I also know that $\Lambda$ is basic $\Leftrightarrow$ $\Lambda/r$ is basic (where $r$ is a Jacobson radical) but cannot see if I can apply this...