If are principal ideal rings, then is also principal ideal ring.

Is this right? Why??

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- Oct 18th 2010, 02:29 AMKaKaprodct of principal ideal rings
If are principal ideal rings, then is also principal ideal ring.

Is this right? Why?? - Oct 18th 2010, 11:18 AMNonCommAlg
yes, if and both have of course. the reason is that every ideal of is in the form for some ideal of and some ideal of .

maybe i don't have to mention that is never a domain. so the result doesn't hold for principal ideal "domains". - Oct 19th 2010, 10:05 PMKaKa
Umm...

Let be an ideal in .

Then how can I find a generator s.t. ??? - Oct 20th 2010, 07:27 AMNonCommAlg
well, assuming that your rings are commutative of course, if and , then