ABSTRACT
In Chapter 1 we saw the role that idempotents in tensor products play in the
concept of a separable extension R ⊆ S of commutative rings (Proposition 1.5). The “separability idempotent” (Definition 2) which appears in S ⊗R S needs to be distinguished from idempotents which are present in the tensor
product because they lie in S or even R. Thus we embark in this chapter and
the next on a study of idempotents in commutative rings. This is very easy
in the case of rings with only finitely many idempotents: such rings are finite
products of rings with no idempotents except 0 and 1. The general case is
more complicated, of course.
