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 -module is tensor a bracket group.
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, , where the intersection is taken over the isolated components 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...
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 , where π⁎(φ) = πφ for each (where maps are acting on the left). On the one hand,...
First, we give complete description of the comultiplication modules over a Dedekind domain. Second, if is the pullback of two local Dedekind domains, then we classify all indecomposable comultiplication -modules and establish a connection between the comultiplication modules and the pure-injective modules over such domains.
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].
Let be a commutative Noetherian ring. It is shown that the finitely generated -module with finite Gorenstein dimension is reflexive if and only if is reflexive for with , and for with . 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 we give a characterization of -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...
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.