Adjoining Idempotents to a Commutative Ring preprint version

Abstract

Everything takes place in the category of commutative unitary rings. For a fixed ring $R$, $\alg{R}$ is the class of $R$-algebras and $\igr{R}$ 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 $A\in \igr{R}$ (both as rings and as $R$-modules). Examples: (1) If $R\sbq A\in \igr{R}$ 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 $A_R$ 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 $\igr{R}$.