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Author: William A. Stein
The ideal  and the ring  next up previous
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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 = RS, 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 RS 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 2U 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 RS 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 RS, 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 RS 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 RS 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).


next up previous
Next: A generalization of parity Up: Parity structures and generating Previous: Refinements of results
William Arthur Stein
1999-10-27