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
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
.
We can identify the elements of
each of the rings *R*_{S} and
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
,
it corresponds to taking the subsets that occur
in both collections. Both rings, however, are free boolean rings
of the same dimension over
,
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
with each
term
an element of
.
A bit of calculation now shows that sending such an idempotent to
the function
and extending linearly
gives a natural isomorphism from *R*_{S} to
.
Finally,
the element
is the sum of those idempotents with an even
number of the terms
equal to 0, so corresponds
in
to the sum of the characteristic functions of the
co-even subsets of *S* (those with even complement in *S*).