(i) Let $R$ be a commutative ring. Define the terms prime ideal and maximal ideal, and show that if an ideal $M$ in $R$ is maximal then $M$ is also prime.

(ii) Let $P$ be a non-trivial prime ideal in the commutative ring $R$ ('non-trivial' meaning that $P \neq\{0\}$ and $P \neq R$ ). If $P$ has finite index as a subgroup of $R$, show that $P$ is also maximal. Give an example to show that this may fail, if the assumption of finite index is omitted. Finally, show that if $R$ is a principal ideal domain, then every non-trivial prime ideal in $R$ is maximal.