Every ideal in a Noetherian ring is an intersection of ideals primary to maximal ideals

Question

I'm reading a lecture note Tight Closure of Huneke and he stated that "Every ideal in a Noetherian ring is
an intersection of ideals primary to maximal ideals". I don't see why this is the case.

I know that every ideal in a Noetherian ring has a primary decomposition into an intersection of ideals primary to prime ideals, but I don't see how I can refine this result.

Here the ring is of prime characteristic, but I don't think that will help.

Answer 1

Let $A$ be a ring, let $I$ be a proper ideal of $A$, let $\mathfrak{m}$ be a maximal ideal of $A$ which contains $I$. Then $I = \bigcap_{n \ge 1} (I + \mathfrak{m}^{n})$ by the Krull intersection theorem, and each $I + \mathfrak{m}^{n}$ is $\mathfrak{m}$-primary.