Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$
I’m really sorry if this is absolutely obvious or absolutely wrong. For example, if $\exists k$ such that $\alpha^k=\alpha^{k+1}=…$ then $\alpha\cap\alpha^2\cap\alpha^3\cap…=\alpha^k$ and the identity I want to prove becomes obvious. I'll be grateful for any help!
I'd like to add that we may suppose $A$ to be noetherian and $\alpha$ the only maximal ideal.
The equality holds because $\alpha \cap \alpha^2 \cap \ldots$ is actually the zero ideal by Krull's intersection theorem. You may want to read 2.3, 2.4 and 2.5 of these notes.