This is regarding Example 1.21 in the textbook $\textit{Algorithmic Methods in Non-Commutative Algebra,J.Bueso, J. Gomez-Torrecillas, A.Verschoren}$. It says: Let us denote by $C = \Bbbk[x]$, where $\Bbbk$ is an algebraically closed field and by $\mathfrak{m}$ the maximal ideal $\langle x \rangle$ of $C$. Let $R = \left[\begin{array}{l}C&\mathfrak{m}\\\mathfrak{m}&C\end{array}\right]$, i.e the subring of $M_2(C)$ consisting of all matrices whose off-diagonal entries belong to $\mathfrak{m}$ . It is fairly easy to see that there are exactly two surjective $\Bbbk$-maps $R \rightarrow \Bbbk$ corresponding to taking the quotient of $R$ by the maximal ideals $M_+ = \left[\begin{array}{l}\mathfrak{m}&\mathfrak{m}\\\mathfrak{m}&C\end{array}\right]$ resp. $M_{-} = \left[\begin{array}{l}C&\mathfrak{m}\\\mathfrak{m}&\mathfrak{m}\end{array}\right]$.
I am not able to see how we can show that there are the only two surjective maps and how these ideals are obtained.