## A basic property of primes

Let $R$ be a commutative ring and let $p,q \in R$ be primes such that $p$ does not divide $q$.. Show that if $a \in R$ such that $p|a$ and $q|a$, then $pq|a$.

Suppose $p|a$ and $q|a$; then there exists $t \in R$ such that $qt = a$. In particular, $p|qt$. Since $p$ is prime in $R$ and $p$ does not divide $q$, $p$ must divide $t$. Say $ps = t$. Then we have $pqs = qt = a$, and so $pq|a$.