Rings Over Which Flat Covers of Finitely Generated Modules are Projective

Amini, Afshin; Ershad, M.; Sharif, Habib

doi:10.1080/00927870802108049

Cited by 10 publications

(18 citation statements)

References 9 publications

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“…If SF is the class strongly flat acts then by Proposition 3.5 we now know the strongly flat covers and SF-covers do not coincide. Conversely, suppose that R is a right reversible and weakly left collapsible submonoid of [1] …”

Section: Lemma 33 (Theorem 27 In [13]) Let S Be a Monoid And S/ρ Amentioning

confidence: 99%

“…We also show that a cyclic S-act S/ρ has a weakly pullback flat cover if and only if [1] ρ contains a right reversible and weakly left collapsible submonoid R such that for all u ∈ [1] ρ , uS ∩ R = ∅.…”

mentioning

confidence: 99%

“…Over the past several decades, the covers of modules have been investigated by many authors and ample results have been obtained (see [1,3,4,12,15]). Covers of acts over monoids are studied in [5,7,8].…”

mentioning

confidence: 99%

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On Flatness Covers of Cyclic Acts Over Monoids

Qiao¹,

Wang²

2011

Glasgow Math. J.

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Abstract. The covers of cyclic acts over monoids were investigated by Mahmoudi and Renshaw (M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum 77 (2008), 325-338) and the authors posed some open problems. In the present paper, we give answers to their problems 1 and 5, and we also give a sufficient and necessary condition that a cyclic act has a weakly pullback flat cover.2000 Mathematics Subject Classification. 20M30 1. Introduction. Throughout this paper, S always stands for a monoid, and N for the set of natural numbers.Over the past several decades, the covers of modules have been investigated by many authors and ample results have been obtained (see [1, 3, 4, 12, 15]). Covers of acts over monoids are studied in [5,7,8]. Further investigations about this field were laid dormant until the recent appearance of [13].Let us recall results and definitions that we shall use below. We refer the reader to [11] for a detailed account of these.A monoid S is said to be right reversible if for any p, q ∈ S there exist u, v ∈ S such that up = vq. A monoid S is said to be weakly left collapsible if for any p, q, r ∈ S with pr = qr there exists u ∈ S such that up = uq.In [2], the acts, now called strongly flat, were introduced: A right S-act A S is strongly flat if the functor A S − preserves pullbacks and equalizers. In the same paper, strongly flat acts were characterized as the acts satisfying two interpolation conditions, later labelled as condition (P) and condition (E):

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Section: Lemma 33 (Theorem 27 In [13]) Let S Be a Monoid And S/ρ Amentioning

confidence: 99%

mentioning

confidence: 99%

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On Flatness Covers of Cyclic Acts Over Monoids

Qiao¹,

Wang²

2011

Glasgow Math. J.

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“…EXAMPLE 14. If R is a left A-perfect ring which is not left perfect (see [2] for such a ring), then R ‫)ގ(‬ is not A-perfect as a left R-module.…”

Section: G = T(g)mentioning

confidence: 99%

“…The following well-known lemma will be used in this paper (see [2,Lemma 3.6]). The following result may be known but we donot have a reference.…”

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confidence: 99%

Almost-Perfect Modules

Aydoğdu

Özcan

2010

Glasgow Math. J.

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Abstract. We call a module M almost perfect if every M-generated flat module is M-projective. Any perfect module is almost perfect. We characterize almost-perfect modules and investigate some of their properties. It is proved that a ring R is a left almost-perfect ring if and only if every finitely generated left R-module is almost perfect. R is left perfect if and only if every (projective) left R-module is almost perfect.

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Almost-Perfect Rings and Modules

Amini

Ershad

2009

Communications in Algebra

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Rings Over Which Flat Covers of Finitely Generated Modules are Projective

Cited by 10 publications

References 9 publications

On Flatness Covers of Cyclic Acts Over Monoids

On Flatness Covers of Cyclic Acts Over Monoids

Almost-Perfect Modules

Almost-Perfect Rings and Modules

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