I have faced some problem while cross with them in localization of ring. $1)$ Give examples of distinct proper ideals $I$ and $J$ of a ring $R$ and a multiplicative subset $S$ of $R$ such that $S^{-1}I=S^{-1}J$.
$2)$ Let $R$ be a commutative ring with $1≠0$. $S$ be the set of all units of $R$. Then show that $f:R→S^{-1}R$ is an isomorphism.
$3)$ Let $R$ be a commutative ring with $1≠0$. Let $x∈R$ and $S=\bigl\{1,x,x^2,x^3,.........\bigr\}$. Then show that $S^{-1}R$ is isomorphic to $R(Y)/(xY-1)$.
After thinking enough I can do only first one by taking $R=Z$ and $S=$ set of all odd integers and $I=14Z$ and $J=6Z$.
Is my answer of the $1$st question correct? Please help me to solve the others.
Hint:
For question $3)$, the isomorphism is with $/(xY-1)$. $R(Y)$ isn't really defined if $R$ is not an integral domain.
This isomorphism can be defined this way: first consider the ring homomorphism: \begin{align} \varphi\colon R[Y]&\longrightarrow R_x\\ Y&\longmapsto\varphi(Y)=\frac1x \end{align} and check $\;\ker\varphi$ is the principal ideal generated by $xY-1$.
Some details:
Let $P(Y)\in R[Y]$ such that $P(\frac1x)=0$ in $R_x$, and consider the canonical image of $P(Y)$ in $R_x[Y]$: since $\frac1x$ is a root, it is divisible by $Y-\frac1x$ in this polynomi