If are principal ideal rings, then is also principal ideal ring. Is this right? Why??
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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".
Umm... Let be an ideal in . Then how can I find a generator s.t. ???
well, assuming that your rings are commutative of course, if and , then
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