Can nonisomorphic indecomposable projective modules contain isomorphic projective submodules?

Question

Let $P_1$ and $P_2$ be nonisomorphic indecomposable projective $R-$modules.

Can they contain isomorphic projective submodules?

The answer would be extremely helpful for my studies of basic algebras.

Answer 1

Yes, this is possible.

For an easy example are you familiar with the path algebra of a quiver? If we take the quiver $Q = 1 \rightarrow 2 \leftarrow 3$ then the indecomposable projective modules for the path algebra $kQ$ are given by the representations

$$P(1) = k \overset{\mathrm{id}}{\rightarrow} k \leftarrow 0,$$ $$P(2) = 0 \rightarrow k \leftarrow 0,$$ $$P(3) = 0 \rightarrow k \overset{\mathrm{id}}{\leftarrow} k.$$

These are pairwise nonisomorphic and $P(2)$ is isomorphic to a submodule of both $P(1)$ and $P(3)$.