Let $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:

(i) $p$ is prime;

(ii) $p$ is irreducible;

(iii) $(p)$ is a maximal ideal of $R$;

(iv) $R /(p)$ is a field;

(v) $R /(p)$ is an Integral Domain.

Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.

Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.

Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.