Maximal immediate extension of a fibre product of valuation domains (Q759803)

Every valuation domain R is the pullback \((=fibre\) product) of \(R_ P\) and R/P over the field \(R_ P/P\). Here P is any prime ideal of R. Moreover, the pure-injective envelope of the R-module R coincides with the R-module structure of any maximal immediate extension of R. In this paper we describe the pure-injective envelope of R in terms of its pullback components \(R_ P\) and R/P. It turns out that the pure- injective envelope of R is the fibre product of the pure-injective envelope of R/P and the pure-injective envelope of a free \(R_ P\)-module F of rank equal to the torsion-free rank of the pure-injective envelope of R/P.