When Self-Injective Rings Are Qf: A Report on a Problem

Faith, Carl; Huynh, Dinh Van

doi:10.1142/s0219498802000070

Cited by 30 publications

(13 citation statements)

References 93 publications

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“…The following corollary solves the problem posed in [11, Remarks before Theorem 2.7] (see also [7,Chapter 4] and [4,Paragraph 9]). Corollary 6.…”

Section: By Lemma 2 and It Makes Sense To Define An Endomorphismmentioning

confidence: 84%

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Every ℵ1-Σ-CS module is Σ-CS

Asensio

2004

Journal of Algebra

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It is proved that if M is a uniform module such that M (ℵ 1 ) is CS, then its endomorphism ring is local. As a consequence, it is shown that every ℵ 1 -Σ-CS module is Σ-CS. This solves open questions asked in [Contemp. Math. 259 (2000) 467, J. Algebra 262 (2003) 194]. On the way to these results, it is also shown that the endomorphism ring of a uniform module M is local provided that every local summand of M (ℵ 0 ) is a direct summand.

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“…The following corollary solves the problem posed in [11, Remarks before Theorem 2.7] (see also [7,Chapter 4] and [4,Paragraph 9]). Corollary 6.…”

Section: By Lemma 2 and It Makes Sense To Define An Endomorphismmentioning

confidence: 84%

“…After these results, there appeared the following natural question, asked in [9,11] (see also [4,Section 9]). Assume that we fix an infinite cardinal number ℵ and we know that M (ℵ) is CS.…”

Section: Introductionmentioning

confidence: 99%

Every ℵ1-Σ-CS module is Σ-CS

Asensio

2004

Journal of Algebra

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“…Namely, it was shown in [6] that both conjectures are true when every cyclic left R-module essentially embeds in a projective module. More partial results on these conjectures can be found, for instance, in [4,5,10,13].…”

Section: Conjecture 2 (Fgf Conjecture) Every Left Fgf Ring Is Quasi-mentioning

confidence: 93%

The FGF Conjecture and the Singular Ideal of a Ring

Gómez-Torrecillas

Asensio

2013

J. Algebra Appl.

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We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings. . 1350025-1 J. Algebra Appl. 2013.12. Downloaded from www.worldscientific.com by UNIVERSITY OF CALGARY on 04/13/15. For personal use only. R R u fr / / R R u E gr / / E 1350025-2 J. Algebra Appl. 2013.12. Downloaded from www.worldscientific.com by UNIVERSITY OF CALGARY on 04/13/15. For personal use only. The FGF Conjecture and the Singular Ideal of a Ringwhere u : R → E is the embedding of R R into its injective envelope.Lemma 1. The assignment f r → g r induces a ring homomorphismProof. We only need to check that Φ is well defined. Assume that g, h ∈ S satisfy that gRemark 2. Let us note that, as J(S) consists of those R-endomorphisms of E having essential kernel, Ker(Φ) is the left singular ideal of R,where l R (r) denotes the left annihilator of r in R. Therefore, Φ induces an injective homomorphism of rings Lemma 3. Let R be a ring, E = E( R R), S = End R (E) and J = J(S). There exists an injective map from the set of isomorphism classes of simple left ideals of R R to the set of isomorphism classes of simple left ideals of S/J.Proof. We will adapt the proof given in [6, Lemma 2.1]. Let us define this map as follows. If C ⊆ Soc( R R) is a simple left ideal, then E( R C) is an indecomposable direct summand of E. Therefore, there exists an idempotent e C ∈ S such that E(C) = Ee C . As E(C) is indecomposable, e C is a primitive idempotent of S and, as eSe = End(Se) is a local ring, we get that Se C is the projective cover of the simple module Se C /Je C ∈ Soc( S/J S/J). We map C → Se C /Je C . Clearly, the above assignments are unique up to isomorphism.

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“…A/Jac(A) has finite length (equivalently, it is semi-simple). Much work has been dedicated to this problem over the years [3-5, 9, 16-21]; we also refer the reader to the recent survey [7], which contains a comprehensive account of the history and known results on QF rings and related topics.…”

Section: Conjecture 11 (Faith) a Left Self-injective Semi-primary Rmentioning

confidence: 99%

On Semi-Primary Self-Injective Rings and Faith's Quasi-Frobenius Conjecture

Iovanov

2014

Proceedings of the Edinburgh Mathematical Society

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A long-standing conjecture of Faith in ring theory states that a left self-injective semi-primary ring A is necessarily a quasi-Frobenius ring. We propose a new method for approaching this conjecture, and prove it under some mild conditions; we show that if the simple A-modules are at most countably generated over a subring of the centre of A, then the conjecture holds. Also, the conjecture holds for K algebras A over sufficiently large fields, i.e. if the cardinality of K is larger than the dimension of the simple A-modules (or of A/Jac(A)). This effectively proves the conjecture in many situations, and we obtain several previously known results on this problem as a consequence.

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When Self-Injective Rings Are Qf: A Report on a Problem

Cited by 30 publications

References 93 publications

Every ℵ1-Σ-CS module is Σ-CS

Every ℵ1-Σ-CS module is Σ-CS

The FGF Conjecture and the Singular Ideal of a Ring

On Semi-Primary Self-Injective Rings and Faith's Quasi-Frobenius Conjecture

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