Let \(R\) be a commutative ring. A proper ideal \(I\subsetneq R\) is primary if for all \(p,q\in R\) with \(pq\in I\), either \(p\in I\) or there exists \(n\geq 1\) such that \(q^n\in I\).
In a commutative ring, every maximal ideal is prime, and every prime ideal is both radical and primary. [@AM18]
In \(k[x,y]\) the ideal \((xy)\) is radical but not primary, since \(x \cdot y \in (xy)\) while \(x \notin (xy)\) and no power of \(y\) lies in \((xy)\). In \(k[x]\) the ideal \((x^2)\) is primary but not radical, since \(x \notin (x^2)\) and \(x^2 \in (x^2)\).
Show that maximal implies prime implies both radical and primary, and that neither radical nor primary implies the other.