The ring of upper triangular matrices as a module over itself $R$ is taken to be the ring of upper $3 \times 3$ matrices with entries in $\mathbb{R}$.
If I view $R$ as a module over itself, are any of its submodules free?
And how can I prove that its submodules are pojective $R$-modules?
Thanks!
 A: None of its proper submodules could be free. Think about it: $R$ is a $6$ $\Bbb R$ dimensional algebra. It could not contain even a single copy of itself properly, considering that a single copy would have dimension at least $6$.
It certainly contains projective submodules, though. For any idempotent $e$, $eR$ is going to be a summand of $R$, and hence a projective module.
Actually triangular matrix rings over fields are hereditary meaning that all their left and right ideals are projective.

The only proofs I'm aware of for this would be a little long to write out. I know a proof appears in section 2 of Lectures on Modules and Rings, and First Course in Noncommutative rings section 25, both books by T.Y. Lam.
The version I like is the one where you show that a ring is right hereditary iff its Jacobson radical is a projective right $R$ module.
A: The algebra you're studying is the path algebra over the quiver
$$
\underset{1}{\bullet}\xrightarrow{\alpha}\underset{2}{\bullet}
\xrightarrow{\beta}\underset{3}{\bullet}
$$
so it's hereditary as all path algebras. You find a proof of this result in a paper by C. M. Ringel, see section 3, page 3.
