Maximal ideals of a subring

Question

What are the maximal ideals of $R_5=\{x\in\mathbb{Q}|x=a/b,5\nmid{b},a,b\in\mathbb{Z} \}\subset\mathbb{Q}$? I am thinking that they are of the form $R_5(x)$ with x$\in\mathbb{Z}$ since if $I$ a maximal ideal then $I=R_5(a/b)$ but $a/b=a*1/b$ and $b,1/b\in R_5$. Is this correct?

Answer 1

That is a step in the right direction, however not all $xR_5$ are maximal ideals, in fact there is just one $x$ satisfying this.

To see this:

Consider intersections of the form $\mathfrak a = \mathfrak m \cap \mathbb Z$, for a maximal ideal $\mathfrak m$ of $R_5$. What can you say about $\mathfrak a$?

Then consider the ideal $\mathfrak a R_5$. Is this prime/maximal? What can you say about the elements not contained in $\mathfrak a R_5$?