The ideal  $\sigma R_{S}$ and the ring  $2^{{\mathcal P}(S)}$

We should like to make a few remarks pointing out another way to think about the ideal mentioned in the proof of Theorem 1. It is straightforward to see that the ring R = R_S, with S of size n, is semisimple and is isomorphic to  $\mbox{\bf F}_2^{n}$ via the set of primitive (central) orthogonal idempotents  $\{\prod_{s \in S} (x_{s} + \epsilon_{s}) \, \vert \, \epsilon_{s} \in \mbox{\bf F}_2\}$. Since the ideals of  $\mbox{\bf F}_2$ are just itself and (0), the elements of the ideal $\sigma R$ are just those elements of R that have support on the coordinates corresponding to the idempotents lying in $\sigma R$. An idempotent  $t = \prod (x_{s} + \epsilon_{s})$ will be in the ideal $\sigma R$ if and only if t equals  $\sigma t = t + \sum_{s \in S} x_{s} t$. This happens exactly if an even number of the terms in the summation equal t (while the rest vanish), which happens if and only if an even number of the field elements  $\epsilon_{s}$ equal 0. To summarize, the elements of $\sigma R$ are the  $\mbox{\bf F}_2$-linear combinations of those idempotents  $\prod (x_{s} + \epsilon_{s})$ that have an even number of the elements  $\epsilon_{s}$ equal to 0, which is the same as having the lowest-degree term be of even degree.

We now point out a relationship between the ring R_S used in our proofs above and a similar boolean ring Halmos discusses in his boolean algebra book [1]. For a set U, he considers the ring 2^U of functions from U to  $\mbox{\bf F}_2$ with coordinatewise addition and multiplication. Notice that the sum of the characteristic functions of two subsets of U is the characteristic function of their symmetric difference, while the product is the characteristic function of their intersection. This construction can, of course, be applied with the power set of S in place of U to obtain the ring  $2^{{\mathcal P}(S)}$. We can identify the elements of each of the rings R_S and  $2^{{\mathcal P}(S)}$ with collections of subsets of S, and in each ring, addition corresponds to taking the symmetric difference of two collections. Multiplication, however, has different interpretations in the two rings. In R_S, it corresponds to taking all unions of a subset from the first collection and a subset from the second collection (counted with appropriate multiplicities modulo 2), while in  $2^{{\mathcal P}(S)}$, it corresponds to taking the subsets that occur in both collections. Both rings, however, are free boolean rings of the same dimension over  $\mbox{\bf F}_2$, and so are isomorphic.

As indicated above, the primitive orthogonal idempotents of the ring R_S are (with our usual notation) all products of the form  $\prod_{i=1}^{n} (x_{i} + \epsilon_{i})$ with each term  $\epsilon_{i}$ an element of  $\mbox{\bf F}_2$. A bit of calculation now shows that sending such an idempotent to the function  $x_{i} \mapsto \epsilon_{i}$ and extending linearly gives a natural isomorphism from R_S to  $2^{{\mathcal P}(S)}$. Finally, the element $\sigma$ is the sum of those idempotents with an even number of the terms  $\epsilon_{i}$ equal to 0, so corresponds in  $2^{{\mathcal P}(S)}$ to the sum of the characteristic functions of the co-even subsets of S (those with even complement in S).