Endomorphism ring of $x \Bbbk\langle x,y \rangle + y \Bbbk\langle x,y \rangle$

Question

Let $R = \Bbbk \langle x,y \rangle$, where $\Bbbk$ is a field. I want to determine $\underline{\text{End}}_R(xR + yR)$, the ring of (not necessarily degree-preserving) graded module homomorphisms of the right ideal $xR + yR$.

I know that we have an inclusion $R \hookrightarrow \underline{\text{End}}_R(xR + yR)$ given by $p \mapsto \phi_p$, where $\phi_p(q) = pq$, and I'm inclined to believe that this is actually an isomorphism, but I can't prove surjectivity. Is this an isomorphism, and if so, how should I go about proving it?

Answer 1

$xR\cap yR=\{0\}$, so your ideal is the direct sum of $xR$ and $yR$, both of which are isomorphic to $R$ as right ideals, so its endomorphism algebra is isomorphic to $M_2(R)$.