When every torsion preradical is a torsion radical

Berg, John E. van den

doi:10.1080/00927879908826771

Cited by 6 publications

(4 citation statements)

References 13 publications

(6 reference statements)

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“…Furthermore, if R is a domain and every ideal of R is idempotent, then every F ∈ R-fil is a left Gabriel topology, or equivalently, every T ∈ R-torsp is a hereditary torsion class [7,Theorem 28]. Now suppose that R is a left chain domain whose only proper nonzero ideal is the Jacobson radical J (R).…”

Section: Proof a Cogenerator For σ [M] Is Given Bymentioning

confidence: 99%

Modules whose hereditary pretorsion classes are closed under products

Berg

Wisbauer

2007

Journal of Pure and Applied Algebra

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A module M is called product closed if every hereditary pretorsion class in σ [M] is closed under products in σ [M]. Every module M which is locally of finite length (every finitely generated submodule of M has finite length) is product closed and every product closed module M is semilocal (M/J (M) is semisimple). Let M ∈ R-Mod be product closed and projective in σ [M]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invariant submodules; (3) M has finite length if M is finitely generated and every hereditary pretorsion class in σ [M] is M-dominated. If the ring R is commutative it is proven that M is product closed if and only if M is locally of finite length. It was shown by Beachy and Blair [2, Proposition 1.4 and Corollary 3.3] that the following three conditions on a ring R with identity are equivalent: (1) every hereditary pretorsion class in R-Mod is closed under arbitrary (and not just finite) direct products, or equivalently, every left topologizing filter on R is closed under arbitrary (and not just finite) intersections; (2) every left R-module M is finitely annihilated, meaning (0 : M) = (0 : X ) for some finite subset X of M; (3) R is left artinian. In this paper we shall attempt to describe those modules M with the property that every hereditary pretorsion class in the Grothendieck category σ [M] is closed under products in σ [M]. One of our main results says that if M is a finitely generated product closed module such that M is projective in σ [M] and every hereditary pretorsion class in σ [M] is M-dominated (meaning, every hereditary pretorsion class in σ [M] is subgenerated by an M-generated module), then M has finite length. This result extends Beachy and Blair's characterization of left artinian rings. Their proof is based on two results due to Beachy [1, Propositions 1 and 5], but the techniques used by Beachy are not easily generalized in a manner useful for our purposes. We have thus had to develop new methods.

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Section: Proof a Cogenerator For σ [M] Is Given Bymentioning

confidence: 99%

Modules whose hereditary pretorsion classes are closed under products

Berg

Wisbauer

2007

Journal of Pure and Applied Algebra

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“…A long-standing open problem is the following: what is the structure of a ring for which every left exact preradical is a radical? Van den Berg showed in [16] that such a ring does not have to be right Noetherian or a right V-ring. Dauns and Zhou showed in [5,Proposition 3.7] that right Noetherian rings whose left exact preradicals are radicals are precisely right QI-rings.…”

Section: When Is I M a Radical?mentioning

confidence: 99%

Rings Characterized via a Class of Left Exact Preradicals

2015

Proceedings of the Edinburgh Mathematical Society

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For two modules M and N, iM(N) stands for the largest submodule of N relative to which M is injective. For any module M, iM: Mod-R → Mod-R thus defines a left exact preradical, and iM(M) is quasi-injective. Classes of ring including strongly prime, semi-Artinian rings and those with no middle class are characterized using this functor: a ring R is semi-simple or right strongly prime if and only if for any right R-module M, iM(R) = R or 0, extending a result of Rubin; R is a right QI-ring if and only if R has the ascending chain condition (a.c.c.) on essential right ideals and iM is a radical for each M ∈ Mod-R (the a.c.c. is not redundant), extending a partial answer of Dauns and Zhou to a long-standing open problem. Also discussed are rings close to those with no middle class.

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“…We show now that β is subgenerated by a direct sum of strongly prime modules. Let I be the set of all proper nonzero ideals of R. By [13,Proposition 27,p5539] every element of I is completely prime. It follows that R/I is a domain for all I ∈ I.…”

Section: Dusemiprime Modulesmentioning

confidence: 99%

“…Notice that the operation ':' defined here is opposite to the multiplication operation introduced in[12],[13] and[11]. Consequently, properties which are prefixed with 'left' in the aforementioned papers, become 'right' in this paper.…”

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Duprime and dusemiprime modules

Berg¹,

Wisbauer

2001

Journal of Pure and Applied Algebra

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A lattice ordered monoid is a structure L; ⊕, 0 L ; ≤ where L; ⊕, 0 L is a monoid, L; ≤ is a lattice and the binary operation ⊕ distributes over finite meets. If M ∈ R-Mod then the set IL M of all hereditary pretorsion classes of σ[M ] is a lattice ordered monoid with binary operation given by α : M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ IL M (the subscript in : M is omitted if σ[M ] = R-Mod). σ[M ] is said to be duprime (resp. dusemiprime) if M ∈ α : M β implies M ∈ α or M ∈ β (resp. M ∈ α : M α implies M ∈ α), for any α, β ∈ IL M. The main results characterize these notions in terms of properties of the subgenerator M. It is shown, for example, that M is duprime (resp. dusemiprime) if M is strongly prime (resp. strongly semiprime). The converse is not true in general, but holds if M is polyform or projective in σ[M ]. The notions duprime and dusemiprime are also investigated in conjunction with finiteness conditions on IL M , such as coatomicity and compactness.

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When every torsion preradical is a torsion radical

Cited by 6 publications

References 13 publications

Modules whose hereditary pretorsion classes are closed under products

Modules whose hereditary pretorsion classes are closed under products

Rings Characterized via a Class of Left Exact Preradicals

Duprime and dusemiprime modules

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