examples of radicals of ideals in commutative rings
Let be a commutative ring. Recall, that ideals in are called coprime iff . It can be shown, that if are coprime, then . Elements are called pairwise coprime iff for . It follows by induction, that for pairwise coprime we have ,
Let be such that
for some prime elements , and assume that are coprime. Denote by
We shall denote by the radical of an ideal .
Proof. ,,” Let . Then we have
and thus . This shows the first inclusion.
,,” Assume that and . Then there is such that . Thus divides . Of course for any we have that divides . Thus divides and since is prime, we obtain that divides . Now for elements and are coprime, thus divides and therefore , which completes the proof.
Remark. If we assume that is a PID (and thus UFD), then the previous proposition gives us the full characterization of radicals of ideals in . In particular an ideal in PID is radical if and only if it is generated by an element of the form , where for elements and are not associated primes.
Examples. Consider ring of integers . Then we have:
|Title||examples of radicals of ideals in commutative rings|
|Date of creation||2013-03-22 19:04:34|
|Last modified on||2013-03-22 19:04:34|
|Last modified by||joking (16130)|