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A class of torsion-free abelian groups characterized by the ranks of their socles

Ulrich F. Albrecht, Anthony Giovannitti, H. Pat Goeters (2002)

Czechoslovak Mathematical Journal

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket R -module is R tensor a bracket group.

Associated primes and primal decomposition of modules over commutative rings

Ahmad Khojali, Reza Naghipour (2009)

Colloquium Mathematicae

Let R be a commutative ring and let M be an R-module. The aim of this paper is to establish an efficient decomposition of a proper submodule N of M as an intersection of primal submodules. We prove the existence of a canonical primal decomposition, N = N ( ) , where the intersection is taken over the isolated components N ( ) of N that are primal submodules having distinct and incomparable adjoint prime ideals . Using this decomposition, we prove that for ∈ Supp(M/N), the submodule N is an intersection of -primal...

Cellular covers of cotorsion-free modules

Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann (2012)

Fundamenta Mathematicae

In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism π : H o m R ( G , G ) H o m R ( G , H ) , where π⁎(φ) = πφ for each φ H o m R ( G , G ) (where maps are acting on the left). On the one hand,...

Comultiplication modules over a pullback of Dedekind domains

Reza Ebrahimi Atani, Shahabaddin Ebrahimi Atani (2009)

Czechoslovak Mathematical Journal

First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if R is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication R -modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.

Divisible ℤ-modules

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

k -torsionless modules with finite Gorenstein dimension

Maryam Salimi, Elham Tavasoli, Siamak Yassemi (2012)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring. It is shown that the finitely generated R -module M with finite Gorenstein dimension is reflexive if and only if M 𝔭 is reflexive for 𝔭 Spec ( R ) with depth ( R 𝔭 ) 1 , and G- dim R 𝔭 ( M 𝔭 ) depth ( R 𝔭 ) - 2 for 𝔭 Spec ( R ) with depth ( R 𝔭 ) 2 . This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n 2 we give a characterization of n -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

Large superdecomposable E(R)-algebras

Laszlo Fuchs, Rüdiger Göbel (2005)

Fundamenta Mathematicae

For many domains R (including all Dedekind domains of characteristic 0 that are not fields or complete discrete valuation domains) we construct arbitrarily large superdecomposable R-algebras A that are at the same time E(R)-algebras. Here "superdecomposable" means that A admits no (directly) indecomposable R-algebra summands ≠ 0 and "E(R)-algebra" refers to the property that every R-endomorphism of the R-module, A is multiplication by an element of, A.

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