# Noetherian ring and primary decomposition result

I'm struggling with the following problem and I would appreciate some help if possible

Let $R$ be Noetherian and let $I,J$ be ideals. Define $(I:J^{\infty}) = \bigcup_{n}(I:J^{n})$.

(a) If $Q$ is primary, prove that $(Q:J^{\infty}) = Q$ for any $J \subset R$ with $J$ not contained in the radical of $Q$.

(b) If $P$ is prime and $I = Q_{1} \cap \ldots \cap Q_{k}$ is a finite intersection of primary ideals, then show that $(I : P^{\infty})$ is the intersections of the $Q_{i}$ for which $P \not \subset P_{i}$.

Thank you!

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(a) If $aJ^n\subseteq Q$, $Q$ primary and $J\nsubseteq \sqrt{Q}$ (therefore $J^n\nsubseteq \sqrt{Q}$), then... – user26857 Oct 17 '12 at 9:11
(b) Apply (a) using that $(Q_1\cap\cdots\cap Q_k:P^{\infty})=(Q_1:P^{\infty})\cap\cdots\cap(Q_k:P^{\infty})$. – user26857 Oct 17 '12 at 9:20
Thanks a lot! I had (b), but I don't know why I couldn't see a) :)); this is what happens when you don't sleep enough – Dquik Oct 18 '12 at 1:46
An alternative definition of primary ideals is the following: $Q$ is primary iff for every ideals $I,J$ with $IJ\subseteq Q$ we have $I\subseteq Q$ or $J\subseteq\sqrt{Q}$ (or viceversa). IN my answer to (a) take $I=(a)$. – user26857 Oct 18 '12 at 8:05