Ring of fractions of $\mathbb Z_6$ I am studying commutative algebra and I have a problem in calculating the fraction rings. For example, it would be a great help someone help me to calculate the ring of fraction  if 
$
S= \{1,3,5\}
$
is a multiplicative subset of $R=\mathbb{Z}_6$.
I know the definition of the ring of fractions, but I can not calculate it.
Thanks for any help.
 A: Well, it's a ring of fractions, so the elements are just that: fractions. The possible fractions are
$$
\frac01, \frac11, \frac21, \frac31, \frac41,\frac51,\frac61,\\ 
\frac03, \frac13, \frac23, \frac33, \frac43,\frac53,\frac63,\\ 
\frac05, \frac15, \frac25, \frac35, \frac45,\frac55,\frac65
$$
Just like the fractions you're used to (rational numbers), some of these are the same. We are allowed to expand fractions using any element of $S$ and it won't change the values of the fractions. For instance, just like you would expect, we have $\frac01 = \frac03 = \frac05$. But what you might not expect is that $\frac23 = \frac{2\cdot 3}{3\cdot 3} = \frac03$. In other words, any fraction with an even numerator is equal to $\frac03$.
In fact, we can expand any fraction so that it has $3$ in the denominator, and what we end up with then are just
$$
\frac03 = \frac23 = \frac43\\
\frac13 = \frac33 = \frac53
$$
So in the end we have only $2$ distinct elements in the final fraction ring. It's not difficult to see that it is isomorphic to $\Bbb Z_2$.
