Zero Ideal is Indecomposable as a Finite Intersection of Primary Ideals on an Infinite Ring of Idempotents

Question

Let R be a ring of idempotents, that is, a ring in which holds $x^2=x, \forall x \in R$. Prove that if R is infinite, then (0) is indecomposable as a finite intersection of primary ideals.
Any hints on this one? I've tried many things, nothing worked...

Answer 1

Here's an outline of an approach:

Step 1: Prove every prime ideal of $R$ has index $2$ (think about the quotient, it will be an integral domain consisting of idempotents).

Step 2: Prove every primary ideal of $R$ must be prime (this can be quick if you know that the radical of a primary ideal is always prime).

Step 3: Suppose we have a decomposition $(0)=\mathfrak q_1\cap\cdots\cap\mathfrak q_n$. Recall we can always replace this with a reduced decomposition (this matters for step 4).

Step 4: Apply chinese remainder theorem to $R=R/(\mathfrak q_1\cap\cdots\cap\mathfrak q_n)$. Combine steps $1$ and $2$ to deduce that $R$ is finite.