Quick question about a ring of fractions

Question

Given a ring $R$ and a multiplicatively closed system $S\subset R$, we define the ring $S^{-1}R=\{\frac rs | r\in R, s\in S\}$. However if I have $R=\Bbb Z$ and $S=\Bbb Z \backslash 3\Bbb Z=\{0,1,2\}$, do the denominators in $S^{-1}R$ contain only $1$ and $2$ or do they contain all the non multiples of $3$ in $\Bbb Z $?

Answer 1

Be careful: $3\mathbb{Z}=\{m\in \mathbb{Z}: m=3k\:\text{for}\:k\in \mathbb{Z}\}.$ $S=\mathbb{Z}\setminus 3\mathbb{Z}$ is indeed a multiplicative system, but actually consists of all integers not divisible by $3$. Then $S^{-1}R$ in your notation is the set of fractions $\frac{r}{s}$ so that $r\in \mathbb{Z}$ and $3\nmid s$. For instance $\frac{3}{2}\in S^{-1}R$ as is $\frac{79}{14}$.

Answer 2

You are misunderstanding what $\mathbb Z\setminus 3\mathbb Z$ means. It is an unfortunate notation in algebra. What you are confusing it with is $\mathbb Z/3\mathbb Z$, which is a quotient rather than a subset of $\mathbb Z$.

$$\mathbb Z\setminus 3\mathbb Z=\{x\in \mathbb Z\mid 3\nmid x\} $$ So the possible denominators are all integers that are not multiples of $3$.

Note for clarity I am using \setminus instead of \backslash.