I'm having some trouble solving these tasks:
Let R be an integral domain, $a \in R-\{0\}$ an element and $S:=\{a^n\}_{n \geq 0}\subset R$
a) Prove that the mapping $R[X]\rightarrow S^{-1}R$ given by $X \rightarrow \frac{1}{a}$ has an isomorphism $R[X]/(aX- 1)\rightarrow S^{-1}R$ induced.
b) Let p be a prime number and consider the ring
$\mathbb{Z}_{(p)}=\{ \frac{x}{y}\in \mathbb{Q }: x,y \in \mathbb{Z} y\neq 0, y\equiv 0 \text{ mod p} \}$
Show that $\mathbb{Z}_{(p)}$ is a principal ideal ring but not a field.
for a) I wanted to solve the exercise using the Isomorphism theorem.
$f:X \rightarrow \frac{1}{a}$ is a surjective homomorphism.
If we prove that $(aX-1)=ker(f)$ we can use the isomorphism theorem and we have solved the exercise, right? But I am having problems showing $(aX-1)=ker(f)$, can someone help me? Is it the right way to solve this problem?
For b) I think, that I have shown correctly that $\mathbb{Z}_{(p)}$ is a principal ideal ring:
We have a principal ideal ring if every ideal $I\subseteq \mathbb{Z}_{(p)}$ is generated by a $d \in \mathbb{Z}_{(p)}$.
since $\mathbb{Z}$ is a principal ideal ring we have that for every $a=\frac{x}{y} \in I$ that $a=\frac{rd}{y}=\frac{r}{y}d$ also it is generated by $(d)$ and $\frac{r}{y} \in \mathbb{Z}_{(p)}$
Have I done it right?
For showing that $\mathbb{Z}_{(p)}$ is not a field I thought to show that it exists al least an element of $\mathbb{Z}_{(p)}$ which is not invertible, but I have no idea on hoe to show this. Can someone help me?