Adjoining idempotents to a commutative ring

Everything takes place in the category of commutative unitary rings. For a fixed ring R, AlgR is the class of R-algebras and IGR the subclass of idempotent generated R-algebras. Following Bezhanishvili et al. and their study of Specker and locally Specker R-algebras, this paper studies the interplay of properties of R and A∈IGR (both as rings and as R-modules). Examples: (1) If R⊆A∈IGR and R is weak Baer (aka p.p. ring) and A is ring essential over R, then A is weak Baer and locally Specker. (2) If R is semiprime and all the idempotents of the complete ring of quotients are adjoined to R to form A, then AR is flat iff R is weak Baer, in which case A is locally Specker. The Pierce sheaf is often used since it is based on idempotents. Properties are examined, old and new, that are true for R iff they are true for all the Pierce stalks. Among the new is the result for f-rings (pure ideals are generated by idempotents): R is an f-ring iff each of its Pierce stalks has no non-trivial pure ideals. This allows the expansion of the known classes of f-rings; f-rings play important roles in IGR.