characterization of primary ideals
Proposition. Let be a commutative ring and an ideal. Then is primary if and only if every zero divisor
in is nilpotent
.
Proof. ,,” Assume, that we have such that is a zero divisor in . In particular and there is , such that
This is if and only if . Thus either or for some . Of course , because and thus . Therefore , which means that is nilpotent in .
,,” Assume that for some we have and . Then
so both and are zero divisors in . By our assumption both are nilpotent, and therefore there is such that . This shows, that and , which completes
the proof.