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Tilting modules over valuation domains
Full text linkThe structure of tilting modules over valuation domains R is investigated. It is proved that the S-divisible modules introduced by Fuchs-Salce are canonical generators for the tilting
torsion classes over valuation domains, assuming V=L and that the cardinality of the pure-injective hull of R is at most the continuum when the tilting generator has uncountable rank
F-divisible modules and tilting modules over Pruefer domains
No full textTilting modules over Prüfer domains are investigated. Tilting torsion classes over these domains correspond bijectively to finitely generated localizing systems of ideals. For each such system F,
a generalized Fuchs divisible module is constructed which generates the corresponding tilting torsion class
On finitely injective modules and locally pure-injective modules over Pruefer domains
Full text linkOver Matlis valuation domains there exist finitely injective modules
which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Pruefer domains which are either almost maximal, or h-local Matlis, finitely injective torsion modules and complete torsionfree locally pure-injective modules correspond to each others under the Matlis equivalence. Almost maximal Pruefer domains are characterized by the property
that every torsionfree complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind
Warfield domains: module theory from linear algebra to commutative algebra through abelian groups
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A general notion of algebraic entropy and the rank-entropy
Full text linkWe give a general definition of a subadditive invariant i of Mod(R), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R-modules, with respect to i. We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show
that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy
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