I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it.
(a) Give an example of a ring $A$ and distinct multiplicative subset $S$, $T$ so that $S^{-1}A=T^{-1}A$.
(b) Prove that for fixed $S$, there is a maximal multiplicative set $T$ with this property, defined by $T=\{t \in A :at\in S \mbox{ for some } a\in A\}$.
Thank you very much.
Hints
(a) Consider $A=\mathbb{Z}$ and $S=\{2^n:n\ge0, n\ne1\}$; then $2\notin S$, but $2$ is invertible in $S^{-1}\mathbb{Z}$.
(b) Prove “$T$ is a multiplicative set and $S^{-1}A=T^{-1}A$”; next prove that this can't hold for any multiplicative set properly containing $T$.