In a Noetherian ring, it is known that every ideal contains a product of prime ideals.
Is there any example of a Noetherian ring in which an ideal is not equal to any product of prime ideals?
This is a natural question,which I didn't find even as an exercise in common references where the above property is stated.
My intuitive answer: consider a suitable subring of the ring of integers in a number field. For example, the ideal $(2,1+\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$; am I right?