# prodct of principal ideal rings

• Oct 18th 2010, 03:29 AM
KaKa
prodct of principal ideal rings
If $A,B$ are principal ideal rings, then $A \times B$ is also principal ideal ring.
Is this right? Why??
• Oct 18th 2010, 12:18 PM
NonCommAlg
yes, if $A$ and $B$ both have $1$ of course. the reason is that every ideal of $A \times B$ is in the form $I \times J$ for some ideal $I$ of $A$ and some ideal $J$ of $B$.

maybe i don't have to mention that $A \times B$ is never a domain. so the result doesn't hold for principal ideal "domains".
• Oct 19th 2010, 11:05 PM
KaKa
Umm...
Let $I=(a) \times (b)$ be an ideal in $A \times B$.
Then how can I find a generator $c$ s.t. $I=(c)$???
• Oct 20th 2010, 08:27 AM
NonCommAlg
well, assuming that your rings are commutative of course, if $I = Aa$ and $J = Bb$, then $I \times J = (A \times B)(a, b).$