$rad(P)$ non-projective induces all submodules of P non-projective?

Question

Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.

Is it right that every non-zero proper submodule of $P$ is not projective?

Answer 1

No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1\to 2, b: 2\to 4, c: 1\to 3, d:3\to 4$) modulo the ideal spanned by $ba-dc$. Then $\operatorname{rad} (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.