Exercise 4, § 1

Let $A$ be an integral domain and let $S$ be a multiplicatively closed subset of $A$ not containing $0$. Let $S^{-1}A$ denote the ring of elements $s^{-1}a$ ($s \in S, a \in A$) in the field of fractions $K$ of $A$ (I, p. 116). a) Show that if $A$ is a principal ideal domain then so is $S^{-1}A$. b) Let $A$ be a principal ideal domain and let $p$ be an irreducible element of $A$; show that the complement $S$ of the ideal $(p)$ is multiplicatively closed ; in the ring $S^{-1}A$ the ideal $pS^{-1}A$ is the unique maximal ideal, and the quotient field $S^{-1}A/pS^{-1}A$ is isomorphic to $A/(p)$.