Local Rings – Quotient Ring Local if Ring is Local

Question

I want to show: If $R$ is local and $I\neq R$ an ideal, then $R/I$ is also local.

We already know: A Ring $R$ is local if and only if $R-R^{\times} = \{r\in R \, | r \notin R^\times \}$ is an ideal. We also know that if $I$ is an maximal Ideal, $R/I$ is a field.

I have no idea how to show that other than using the above Lemma.

Answer 1

The easiest characterization of local rings here is that a ring is local if and only if it has exactly one maximal ideal. Let $M$ be the maximal ideal of $R$ and $f:R\to R/I$ the surjection. If $R/I$ has a maximal ideal $M'\neq f(M) $, then $f^{-1}(M')\neq M$ is a maximal ideal of $R$, which is a contradiction.