Homomorphic image of local ring is a local ring

Question

If $\varphi:R\longrightarrow S$ is a surjective ring homomorphism from a local ring $R$ with maximal ideal $\mathfrak m$ into a non-zero ring $S$, how can I prove that $S$ is a local ring ?

I wanted to prove that $\varphi(\mathfrak m)$ is the only maximal ideal of $S$, but I don’t seem to make much progress. Could you please help me out ?

Answer 1

Hint: If $J$ is an ideal in $S$ then $\phi^{-1}(J)$ is an ideal in $R$ and hence $\phi^{-1}(J) \subseteq m$.

Use surjectivity to show that $J \subseteq \phi(m)$.