Exercise A0.V.1.13 :: monoid.free

Let $n$ be a non-zero integer. Then $n$ is irreducible (in $\mathbb{Z}$ ) if and only if $n$ is prime.

Suppose $n$ is an irreducible non-zero integer and $n \mid mk.$ Thus, there exists $q \in \mathbb{Z}$ such that $qn = mk.$

Let $d = \gcd(n, k).$ Since $d \mid n$ and $d \mid k,$ there exists $c \in \mathbb{Z}$ such that $cd = n.$

Since $n$ is irreducible, either $c$ is a unit or $d$ is a unit.

Suppose that $c$ is a unit. In this case $n \mid d$ and hence $n \mid k,$ and we’re done.

Suppose instead that $d$ is a unit. In this case, there exist $u, v \in \mathbb{Z}$ such that $nu + kv = 1.$

Hence,

\begin{align*} m & = m(nu + kv) \\ & = mnu + mkv \\ & = mnu + qnv \\ & = n(mu + qv). \end{align*}

And in this case $n \mid m.$

The converse follows by lemma 1.7.

$\blacksquare$

monoid.free