Show that every prime ideal is primary.
Solution
Let \(\mathfrak p\) be a prime ideal. If \[ ab\in\mathfrak p, \] then primality gives \[ a\in\mathfrak p\quad\text{or}\quad b\in\mathfrak p. \] In the second case \(b^1\in\mathfrak p\). Thus whenever \(ab\in\mathfrak p\), either \(a\in\mathfrak p\) or \(b^n\in\mathfrak p\) for some \(n\ge1\), which is exactly the definition of a primary ideal.
Therefore every prime ideal is primary.