A graded ring $R$ is graded-local iff $R_0$ is a local ring?

I think there is an elementary proof, but let's check it.

I'm relying on:

1. There always exist maximal proper homogeneous right ideals (by the usual Zorn's Lemma argument.)

2. The sum of two homogeneous right ideals is again homogeneous.

Suppose $M$ and $N$ are distinct maximal homogeneous right ideals. Then $M+N=R$, and there exists $m+n=1$ with $m\in M$ and $n\in N$. Because of the grading, the grade zero parts must be such that $m_0+n_0=1$, and because $M$ and $N$ are both proper and homogeneous, neither $m_0$ nor $n_0$ can be units of $R_0$. This implies $R_0$ is not local.

By contrapositive then, we have shown if $R_0$ is local, then $R$ is graded local.