This paper continues
earlier investigations into the structure of completions of a (von Neumann)
regular ring R with respect to pseudorank functions, and into the connections
between the ringtheoretic structure of such completions and the geometric
structure of the compact convex set P(R) of all pseudorank functions on
R. In particular, earlier results on the completion of R with respect to a
single N ∈ P(R) are extended to completions with respect to any nonempty
subset X ⊆ P(R). Completions in this generality are proved to be regular
and selfinjective by reducing to the case of a single pseudorank function,
using a theorem that the lattice of σconvex faces of P(R) forms a complete
Boolean algebra. Given a completion R with respect to some X ⊆ P(R), it is
shown that the Boolean algebra of central idempotents of R is naturally
isomorphic to the lattice of those σconvex faces of P(R) which are contained
in the σconvex face generated by X. Consequently, conditions on X are
obtained which tell when R is a direct product of simple rings, and how
many simple ring direct factors R must have. Also, it is shown that the
Xcompletion of R contains a natural copy of the completion with respect to any
subset of X, so in particular the P(R)completion of R contains copies of
all the Xcompletions of R. The final section investigates the question of
when a regular selfinjective ring is complete with respect to some family of
pseudorank functions. It is proved that a regular, right and left selfinjective ring
R is complete with respect to a family X ⊆ P(R) provided only that the
Boolean algebra of central idempotents of R is complete with respect to
X.
