Field of fractions of $\mathbb Z[X]/(2X-1)$ 
What is the field of fractions of $\mathbb Z[X]/(2X-1)$?

$\mathbb Z[X]$ is the polynomial ring over $\mathbb Z$ and $(2X - 1)$ a (maximal?) ideal of the polynomial ring.
I assume that the answer is $\mathbb Q$. But I don't have any intuition. Is there an intuitive way to show this equality?

${\mathbb Z}[X]/(2X - 1) \cong {\mathbb Z}[\frac{1}{2}]$, so, indeed, its field of fractions is ${\mathbb Q}$.
(And, by the way, $(2X - 1)$ is not a maximal ideal: ${\mathbb Z}[\frac{1}{2}$] is not a field.)
