Abstract
Doering and Lequain in 1999 introduced a weak approximation theorem for dependent valuation rings and they proved that every finitely generated lattice-ordered group can be realized as the group of divisibility of a semilocal Bézout overring of a polynomial ring over a field k in infinitely many variables, where each of the valuation rings appearing in the finite intersection has residue field k. Moreover, they proved that every semilocal lattice-ordered group admits a lexico-cardinal decomposition form. In this work, we focus on realizing the semilocal -group over a polynomial ring in finitely many variables. We prove that every semilocal lattice-ordered group having a finite rational rank can be realized as the group of divisibility of a Bézout overring of
up to lexico-cardinal decomposition, where k is a field and
are indeterminates over k and n depends on the group. As a corollary, we prove that every semilocal
-group either finitely generated or divisible with finite rational rank is realizable over
where each of the valuation rings appearing in the finite intersection has residue field k.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgements
I would like to express the deepest gratitude to Bruce Olberding for suggesting this topic. I thank him for the helpful discussions and comments that made to achieve the aim of this article.