Let $R$ be the ring of continuous functions on $[0,1]$ under pointwise addition and multiplication, $c \in [0,1]$, and $M_c$ the ideal defined by the set of $f\in R$ that vanish at $c$.
It is true that $M_c$ is a maximal ideal (I've seen several proofs), but I am having trouble explicitly identifying the inverse of the nonzero elements of $R/M_c$.
Suppressing the dependence on $c$, the element $(M + f)$ has inverse $(M + 1/f)$ only if $f \notin M$ and if $f$ is non-zero on all of $[0,1]$. We only know it's nonzero at $c$, and I don't know how to show if $f$ is not in $M$, then it's nonzero on all of $[0,1]$. Thanks!