Decomposition of comodules

García, Josefa M.; Jara, Pascual; Merino, Luis

doi:10.1080/00927879908826529

Cited by 4 publications

(5 citation statements)

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“…), if A is a noetherian commutative ring, M an Amodule, and E(M ) its injective hull, for any complemented distributive submodule N ⊆ M we have that E(N ) ⊆ E(M ) is distributive. This result does not necessarily hold in a non-commutative framework as the following example shows, see [4].…”

mentioning

confidence: 95%

Lattice decomposition of modules

García¹,

Jara²,

Merino³

2021

Preprint

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The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module M produce these decompositions: the lattice decompositions. In a first étage this can be done using endomorphisms of M , which produce a decomposition of the ring End R (M ) as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module M has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, Supp(M ), of M ; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category σ[M ], the smallest Grothendieck subcategory of Mod − R containing M .

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mentioning

confidence: 95%

Lattice decomposition of modules

García¹,

Jara²,

Merino³

2021

Preprint

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“…In the more general correspondence theorem 3.5, the C(M ) are replaced by Tr(M, C). For coalgebras over fields, C(M ) and Tr(M, C) coincide and 3.5 yields [5, 1.3d], [4,Proposition 7], and [7,Lemma 1.8]. Notice that in [5] closed subcategories in σ[ C * C] are called pseudovarieties.…”

Section: C-comodules and Cmentioning

confidence: 99%

“…These techniques are also used in Zhang [13]. We refer to García-Jara-Merino [3,4] for a detailed description of the corresponding constructions.…”

Section: C-comodules and Cmentioning

confidence: 99%

“…These are Grothendieck categories (remember that we assume R C to be flat). The close connection between comodules and modules is based on the following facts which are proved in [12,Section 3,4].…”

Section: Coalgebras and Comodulesmentioning

confidence: 99%

“…This is the smallest Such decompositions of σ[M ] were investigated in Vanaja [10] and related constructions are considered in García-Jara-Merino [4], Nǎstǎses-cu-Torrecillas [8] and Green [5].…”

Section: Introductionmentioning

confidence: 99%

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Decompositions of modules and comodules

Wisbauer¹

2000

Algebra and Its Applications

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It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable subrings A = A 1 ⊕ · · · ⊕ A n such that the A i -Mod are indecomposable module categories. Similarly any coalgebra C over a field can be written as a direct sum of indecomposable subcoalgebras C = I C i such that the categories of C i -comodules are indecomposable. In this paper a decomposition theorem for closed subcategories of a module category is proved which implies both results mentioned above as special cases. Moreover it extends the decomposition of coalgebras over fields to coalgebras over noetherian (QF) rings.

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