This paper continues
earlier investigations into the structure of completions of a (von Neumann)
regular ring R with respect to pseudo-rank functions, and into the connections
between the ring-theoretic structure of such completions and the geometric
structure of the compact convex set P(R) of all pseudo-rank 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 self-injective by reducing to the case of a single pseudo-rank function,
using a theorem that the lattice of σ-convex 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 σ-convex faces of P(R) which are contained
in the σ-convex 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
X-completion 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 X-completions of R. The final section investigates the question of
when a regular self-injective ring is complete with respect to some family of
pseudo-rank functions. It is proved that a regular, right and left self-injective 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.
