A non-PID

#ring_theory

Let $R$ be an integral domain. Suppose that $a$ and $b$ are non-associate irreducible elements in $R$ , and the ideal $(a, b)$ generated by $a$ and $b$ is a proper ideal. Show that $R$ is not a principal ideal domain (PID).

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Let $R$ be an integral domain. Suppose that $a$ and $b$ are non-associate irreducible elements in $R$, and the ideal $(a,b)$ generated by $a$ and $b$ is a proper ideal. Show that $R$ is not a principal ideal domain (PID).