# prodct of principal ideal rings

• Oct 18th 2010, 02:29 AM
KaKa
prodct of principal ideal rings
If $\displaystyle A,B$ are principal ideal rings, then $\displaystyle A \times B$ is also principal ideal ring.
Is this right? Why??
• Oct 18th 2010, 11:18 AM
NonCommAlg
yes, if $\displaystyle A$ and $\displaystyle B$ both have $\displaystyle 1$ of course. the reason is that every ideal of $\displaystyle A \times B$ is in the form $\displaystyle I \times J$ for some ideal $\displaystyle I$ of $\displaystyle A$ and some ideal $\displaystyle J$ of $\displaystyle B$.

maybe i don't have to mention that $\displaystyle A \times B$ is never a domain. so the result doesn't hold for principal ideal "domains".
• Oct 19th 2010, 10:05 PM
KaKa
Umm...
Let $\displaystyle I=(a) \times (b)$ be an ideal in $\displaystyle A \times B$.
Then how can I find a generator $\displaystyle c$ s.t. $\displaystyle I=(c)$???
• Oct 20th 2010, 07:27 AM
NonCommAlg
well, assuming that your rings are commutative of course, if $\displaystyle I = Aa$ and $\displaystyle J = Bb$, then $\displaystyle I \times J = (A \times B)(a, b).$