We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prufer (in particular Bezout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For B{e}zout domains these conditions are also necessary.
Let B be a commutative Bezout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes of modules over the localizations of B by the maximal ideals of B, and on the other hand, of the constructible subsets of MSpec(B). When B has good factorization, it allows us to derive decidability results for the class B-modules, in particular when B is the ring of algebraic integers or its intersection with real numbers or p-adic numbers.
We study the R-torsionfree part of the Ziegler spectrum of an order Lambda over a Dedekind domain R. We underline and comment on the role of lattices over Lambda. We describe the torsionfree part of the spectrum when Lambda is of finite lattice representation type.
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the algorithms have been used to perform extensive computer experiments.
Lorna Gregory
,Sonia LInnocente
,Gena Puninski
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(2017)
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"Decidability of the theory of modules over Prufer domains with infinite residue fields"
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Lorna Gregory
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