# Tagged: product of ideals

## 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 $g(x)\in J$.

## Problem 174

Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:

(a) The ideal $P$ is a prime ideal.

(b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.