Boolean ring. Representation as direct product? I am reading Atiyah, Macdonald's book "Commutative algebra".
There is an exercise, which states that that if a ring with identity has idempotent element $\ne 0,1$, then the ring is a direct product of 2 other rings (for the proof if $x$ is this idempotent, consider $A = xA \oplus (1-x)A$). In this paper it is proved that a boolean ring with identity is isomorphic to a subdirect product of the copies of $\mathbb{Z}/2\mathbb{Z}$.
My question is: am I right that the finite boolean ring is in fact isomorphic to direct product of the copies of $\mathbb{Z}/2\mathbb{Z}$?
Here is my proof:
consider a boolean algebra $A$, all it's elements are idempotents, so if it is not $\mathbb{Z}/2\mathbb{Z}$, there is decomposition of this ring into $2$ rings (they are also boolean, since homomorphic image of boolean ring is obviously boolean), we proceed with them similarly, until we obtain a decomposition of $A$ into a product of copies of $\mathbb{Z}/2\mathbb{Z}$.
If the statement I made is false, please, say where did I make a mistake.
Thank you very much!
 A: It is correct that any finite boolean ring is a finite product of copies of $\mathbb{F}_2$, and in that case your proof works. However, the infinite boolean rings are far more interesting! For example, inside $\mathbb{F}_2^{\mathbb{N}}$, you can consider the subring generated by $e_1,e_2,\dotsc$, i.e. the vector space generated by $1,e_1,e_2,\dotsc$. It consists of those sequences whose support is finite or cofinite.
More generally, for every totally disconnected compact hausdorff space $X$, aka Stone space, the ring $C(X,\mathbb{F}_2)$ of continuous functions $X \to \mathbb{F}_2$ is a boolean ring, and the famous Stone duality asserts that $X \mapsto C(X,\mathbb{F}_2)$ establishes an anti-equivalence between the category of Stone spaces and the category of boolean rings. The Stone-Čech compactifications of discrete spaces correspond to products of $\mathbb{F}_2$ (in particular, the finite discrete spaces correspond to finite products of $\mathbb{F}_2$), but there are of course more Stone spaces resp. boolean rings. The ring mentioned above corresponds to $\mathbb{N} \cup \{\infty\}$, the one-point compactification of $\mathbb{N}$.
