English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove a transfer theorem which, when combined with theJaffard-Kaplansky-Ohm Theorem, allows results in model theory of modules overB\'ezout domains to be translated into results over Pr\"ufer domains via theirvalue groups. Extending work of Puninski and Toffalori, we show that theextended positive cone of the value group of a Pr\"ufer domain has m-dimensionif and only if its lattice of pp-$1$-formulae has breadth (equivalently width)and that these dimensions are equal. Further, we show that the existence ofthese dimensions is equivalent to the lattice of pp-$1$-formulae havingm-dimension (and hence to its Ziegler spectrum having Cantor-Bendixson rank)and the non-existence of superdecomposable pure-injective modules. Finally, wegive a best possible upper bound for the m-dimension of the pp-$1$-lattice of aPr\"ufer domain in terms of the m-dimension of the extended positive cone ofits value group and show that all ordinals which are not of the form$\lambda+1$ for $\lambda$ a limit ordinal occur as the m-dimension of thepp-$1$-lattice of a B\'ezout domain.

Dimensions on Lattice Ordered Abelian Groups and Model Theory of Modules over Pr\"ufer Domains