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!