The definition of a primary ideal $P$ is : $ x y\in P $implies either $x \in P $or $y^n \in P$ for some$ n > 0$.
My question is, why is it $x$ and not $x^n$ in the definition? Replacing $x$ by $x^n$ doesn't make a difference, does it?
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I feel like the accepted answer here might be of some help -- https://math.stackexchange.com/questions/264879/primary-ideals-confusion-with-definition – PrincessEev Sep 21 '19 at 03:07
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If you are bothered by the "lack of symmetry" in this definition of primary ideals, then use this one: $Q$ is primary iff every zerodivisor in $R/Q$ is nilpotent. (Let $P$ stand for prime ideals!) – user26857 Sep 21 '19 at 10:09
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https://math.stackexchange.com/q/728363/29335 also – rschwieb Sep 21 '19 at 11:26