TITLE: Definition of primary ideal question
QUESTION [7 upvotes]: A primary ideal (in a commutative ring with unity) is an ideal $J$ for which if $ab\in J$, then either $a\in J$ or $b^n\in J$ for some integer $n\geq 1$. So it also implies (due to commutativity) that if $ab\in J$, then $a^m,b^n\in J$ for some integers $m,n\geq 1$.
Wouldn't the latter be a nicer definition for a primary ideal? Why is it not used?

REPLY [3 votes]: Your criterion is not correct as stated: if $a \in J$, it need not be the case that some power of $b$ is in $J$.
Another definition is that zerodivisors in $R/J$ are nilpotent. If $R$ is Noetherian, then this is equivalent to saying that $J$ has exactly one associated prime. This is what is really important, and actually used when primary hypotheses are needed.