Ring of fractions in $\mathbb{Z}/35\mathbb{Z}$

Question

How can I determine $S^{-1}(\mathbb{Z}/35\mathbb{Z})$, where $S$ consists of of all elements of $\mathbb{Z}/35\mathbb{Z}$ except $0,5,10,15,20,25,$ and $30$?

Answer 1

Hint $\ $ In $\rm\:R = \Bbb Z/35,\:$ elements coprime to $\,5,7,\:$ i.e. not in $\rm\,5R\,$ or $\rm\,7R,\,$ are already units. So the only elements of $\rm\:S = R\backslash5R\:$ not already units are $\rm\:7R\backslash0\, =\, 7,14,21,28,\:$ all $\rm\, = 7u,\:$ for unit $\rm\,u.\:$ Hence inverting all elements of $\rm\:S\:$ is equivalent to inverting the single element $\rm\:7.\:$ Therefore $\rm\ S^{-1}R \cong R[x]/(7x\!-\!1) \cong \Bbb Z[x]/(\color{#0a0}{35,7x\!-\!1}) \cong \Bbb Z[x]/(\color{#c00}5,2x\!-\!1)\cong \Bbb Z[x]/(5,x\!-\!3)\cong \Bbb Z/5,\:$ because we can cancel $\,7,\,$ i.e. $\rm\ \color{#0a0}{7x\equiv1,\ 35\equiv 0}\:\Rightarrow\: 0 \equiv 35x \equiv 5(\color{#0a0}{7x})\equiv \color{#c00}5.$

Remark $\ $ One could, instead, use general properties of localization (its universal property, characterization of which elements are killed), but I presume that you have not yet studied these general properties (else the exercise would probably be straightforward).