ABSTRACT
As we noted in Section 2.1 of Chapter 2, the presence of idempotents sig-
nificantly complicates the Galois theory of separable projective algebras over
commutative rings. Galois theory is not unique in this respect. A similar thing
happens in the theory of commutative rings which are regular in the sense of
von Neumann: that is, rings where every element is an idempotent times a
unit. An example is a product, finite or infinite, of fields. Since commutative
von Neumann regular rings differ from fields only in that they have idempo-
tents, their theory should be much like that of fields. An instructive case to
consider is the commutative von Neumann regular rings C(X,Z/Z2) of con-
tinuous Z/Z2 valued functions on a profinite space X . That these are commu-
tative von Neumann regular rings follows from the fact that every element is
idempotent (so the only unit is 1). Commutative rings in which every element
is idempotent are called Boolean rings. The Stone Representation Theorem
shows that any such ring R is the ring of continuous Z/Z2 valued functions
on its maximal ideal space X , and that the latter is profinite.
