Let $A$ be a finite dimensional $K$-algebra where $K$ is an algebraic ally closed field.
Since $A$ finite dimensional, the right $A-$module $A_A=e_1A\oplus e_2A\oplus\cdots \oplus e_nA$ is an indecomposable submodules decomposition, where $\{e_1,e_2,\cdots,e_n\}$ is a complete set of primitive orthogonal idempotents.
We also assume that $A$ is basic, it is $e_iA \ncong e_jA$, if $i\ne j$.
Then we have the following results.
Every simple right $A$-module is isomorphic to one of the modules $$S(1)=e_1A/\mathrm{rad}e_1A,\cdots, S(n)=e_nA/\mathrm{rad}e_nA .$$
Every indecomposable projective right $A$-module is isomorphic to one of the modules $$P(1)=e_1A,\cdots, P(n)=e_nA.$$
Every indecomposable invective right $A$-module is isomorphic to one of the modules $$I(1)=D(Ae_1)\cong E(S(1)),\cdots, I(n)=D(Ae_n)\cong E(S(n)) ,$$ where $D(-)=\mathrm{Hom}_K(-,K)$, $E(M)$ is the injective envelope of $M$ and $P(M)$ is a projective cover of $M$.
Here is my question.
By the Schur lemma, we can get $\mathrm{Hom}_A(S(1),S(1))\cong ~K$.
I wonder if I can get $\mathrm{Hom}_A(P(1),S(1))\cong \mathrm{Hom}_A(S(1),S(1))\cong~\mathrm{Hom}_A(S(1),I(1))\cong~K$ as a $K-$ vector space?
Any help and references are greatly appreciated.
Thanks!