# Definition of primary ideal question

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?

Your "it implies" is wrong, it is possible that $ab \in J$ and $a \in J$ but $b \notin J$.

For example any prime ideal is primary, but it doesn't necesary satisfy your condition....

Added: P.S. I think the equivalent condition you seek is the following: whenever $ab \in J$ we either ( have one of $a$ or $b$ in J ) OR ( $a^m,b^n\in J$ for some integers $m,n$ .)

But then, the standard definition is much cleaner.

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.

• @rschwieb: You're right, this is weaker. I've just decided to remove it altogether – zcn Jan 15 '14 at 4:04
• I went ahead and did the same with the comment :) I remember following the same line of thought a long time ago! – rschwieb Jan 15 '14 at 4:06