(a) Consider the four following types of rings: Principal Ideal Domains, Integral Domains, Fields, and Unique Factorisation Domains. Arrange them in the form $A \Longrightarrow$ $B \Longrightarrow C \Longrightarrow D$ (where $A \Longrightarrow B$ means if a ring is of type $A$ then it is of type $B$ )

Prove that these implications hold. [You may assume that irreducibles in a Principal Ideal Domain are prime.] Provide examples, with brief justification, to show that these implications cannot be reversed.

(b) Let $R$ be a ring with ideals $I$ and $J$ satisfying $I \subseteq J$. Define $K$ to be the set $\{r \in R: r J \subseteq I\}$. Prove that $K$ is an ideal of $R$. If $J$ and $K$ are principal, prove that $I$ is principal.