## Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements

## Problem 623

Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$.

**(a)** Prove that $IJ=(x, 6)$.

**(b)** Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$.