Generation of preradicals

Bican, Ladislav; Jambor, Pavel; Kepka, Tomáš; Němec, Petr

doi:10.21136/cmj.1977.101453

Cited by 9 publications

(14 citation statements)

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“…(ii) M is self-F -split and every direct summand of M which contains F is fully invariant. (2) The following are equivalent:…”

Section: F -Split Objectsmentioning

confidence: 99%

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Split objects with respect to a fully invariant short exact sequence in abelian categories

Crivei¹,

Tütüncü²,

Tribak³

2018

Preprint

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We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to direct sums and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module and comodule categories.

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“…(ii) M is self-F -split and every direct summand of M which contains F is fully invariant. (2) The following are equivalent:…”

Section: F -Split Objectsmentioning

confidence: 99%

“…Then one may define a preradical r −1 of A op by r −1 (A) = A/r(A) for every object A of A. Recall that r is called hereditary if r is a left exact functor, and cohereditary if r −1 is a right exact functor (e.g., see [2]).…”

Section: F -Split Objectsmentioning

confidence: 99%

Split objects with respect to a fully invariant short exact sequence in abelian categories

Crivei¹,

Tütüncü²,

Tribak³

2018

Preprint

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“…Let X = {X i } i∈I be a nonempty collection of pairwise comaximal ideals of R, and let Y be a generating set for M . Then: (1) for each y ∈ Y , there exists a finite subset…”

Section: Decomposition Theoremmentioning

confidence: 99%

“…Theorem 2.7. Let {X i } i∈I be a family of pairwise comaximal ideals of R. Then M = ⊕ i∈I ℓ M (X i ) and each ℓ M (X i ) = 0 if and only if (1) for every m ∈ M , there exists a nonempty finite subset…”

Section: Decomposition Theoremmentioning

confidence: 99%

Module Decompositions Using Pairwise Comaximal Ideals

Birkenmeier

Ryan

2016

Communications in Algebra

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In this paper we show that for a given set of pairwise comaximal ideals {X i } i∈I in a ring R with unity and any right R-module M with generating set Y and C(X i ) = k∈N ℓ M (X k i ), M = ⊕ i∈I C(X i ) if and only if for every y ∈ Y there exists a nonempty finite subset J ⊆ I and positive integers k j such that j∈J X k j i ⊆ r R (yR). We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ACC on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.2010 Mathematics Subject Classification. 16D70.

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“…Then (T(X ), F(X )) is a hereditary torsion theory (see [BKN,II1.3]), the smallest hereditary torsion theory such that all X -modules are torsion. The torsion radical is τ X (M ) = trace(T(X ), M ) and trace(X , M ) τ X (M ) holds.…”

Section: Dual Goldie Torsion Theorymentioning

confidence: 99%