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

Abstract:

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