Ideals in a Von-Neumann regular ring

Question

I want to show the following result directly:

Let R be a regular ring then every ideal is the intersection of maximal ideals containing it.

I know that every ideal of a regular ring is a z-ideal (semiprime), which are intersection of prime ideals, but every prime ideal in a regular ring is maximal and thus the above result follows. See G.Mason z-ideals and prime ideals. But I do not want to use the above direction of proof. Help me get a direct proof.

Answer 1

Suppose $R$ is VNR and let $I$ be a proper ideal.

Then $R/I$ is VNR too.

VNR rings are semiprimitive, so the zero ideal of $R/I$ is the intersection of maximal ideals of $R/I$.

By correspondence, $I$ is the intersection of maximal ideals of $R$ containing $I$.

QED