Embeddings of the maximal quotient ring of a von Neumann regular ring into its completion with respect to a rank function

Abstract

Let R R be a commutative von Neumann regular ring. A pseudorank function on R R is characterized by a probability measure on the spectrum of R R . We exhibit two commutative von Neumann regular rings R R and R ′ R’ with the same spectrum S S , and two rank functions N N and N ′ N’ on R R and R ′ R’ , corresponding to the same probability measure on S S , and such that the maximal right quotient ring Q max ( R ) {Q_{\max }}(R) embeds in the N N -completion R ¯ \bar R of R R while Q max ( R ′ ) {Q_{\max }}(R’) does not embed in R ′ ¯ \overline {R’} .