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.
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 $\Bbb Z_2$.