Strongly Flat and Condition (P) Covers of Acts Over Monoids

Khosravi, Roghaieh; Ershad, M.; Sedaghatjoo, Mojtaba

doi:10.1080/00927870903390660

Cited by 11 publications

(16 citation statements)

References 8 publications

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“…Proof. As pointed out in [4], it is clear that the trivial left S-act has a free cover if and only if S is a group, and groups satisfy Condition (A). Moreover, if S is a group then S is a free cover of S/ρ via the natural S-morphism, for any left congruence ρ.…”

Section: Corollary 53 [4 Corollary 23]mentioning

confidence: 91%

“…The following lemma gives a number of alternative characterisations of Condition (A), taken from [5,2] and [4], with the exception of (v), which clearly follows from the equivalence of its predecessors.…”

Section: Condition (A)mentioning

confidence: 99%

“…In this case we say A is a cover for B (more properly, (A, θ) is a cover for B). If C is a class of left S-acts then S is said to be left C-perfect if every left S-act has a C-cover, that is, a cover lying in C [4]. A left perfect monoid is one which is left Pr-perfect, where Pr is the class of projectives.…”

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confidence: 99%

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COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

Bailey¹,

Gould²,

Hartmann³

et al. 2014

Glasgow Math. J.

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A monoid S satisfies Condition (A) if every locally cyclic left S-act is cyclic. This condition first arose in Isbell's work on left perfect monoids, that is, monoids such that every left S-act has a projective cover. Isbell showed that S is left perfect if and only if every cyclic left S-act has a projective cover and Condition (A) holds. Fountain built on Isbell's work to show that S is left perfect if and only if it satisfies Condition (A) together with the descending chain condition on principal right ideals, MR. We note that a ring is left perfect (with an analogous definition) if and only if it satisfies MR. The appearance of Condition (A) in this context is, therefore, monoid specific. Condition (A) has a number of alternative characterisations, in particular, it is equivalent to the ascending chain condition on cyclic subacts of any left S-act. In spite of this, it remains somewhat esoteric. The first aim of this paper is to investigate the preservation of Condition (A) under basic semigroup-theoretic constructions. Recently, Khosravi, Ershad and Sedaghatjoo have shown that every left S-act has a strongly flat or Condition (P) cover if and only if every cyclic left S-act has such a cover and Condition (A) holds. Here we find a range of classes of S-acts $\mathcal{C}$ such that every left S-act has a cover from $\mathcal{C}$ if and only if every cyclic left S-act does and Condition (A) holds. In doing so we find a further characterisation of Condition (A) purely in terms of the existence of covers of a certain kind. Finally, we make some observations concerning left perfect monoids and investigate a class of monoids close to being left perfect, which we name left$\mathcal{IP}$a-perfect.

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Section: Corollary 53 [4 Corollary 23]mentioning

confidence: 91%

Section: Condition (A)mentioning

confidence: 99%

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COVERS FOR S-ACTS AND CONDITION (A) FOR A MONOID S

Bailey¹,

Gould²,

Hartmann³

et al. 2014

Glasgow Math. J.

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“…11 Let S be a monoid and let ψ : X → Y be an S−epimorphism in which X satisfies condition (E). Then Y satisfies condition (E) if and only if ψ is 1−pure.…”

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confidence: 99%

Covers of Acts Over Monoids and Pure Epimorphisms

Bailey

Renshaw

2014

Proceedings of the Edinburgh Mathematical Society

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In 2001 Enochs' celebrated flat cover conjecture was finally proven and the proofs (two different proofs were presented in the same paper [4]) have since generated a great deal of interest among researchers. In particular the results have been recast in a number of other categories and in particular for additive categories (see for example [2], [3], [22] and [23]). In 2008, Mahmoudi and Renshaw considered a similar problem for acts over monoids but used a slightly different definition of cover. They proved that in general their definition was not equivalent to Enochs', except in the projective case, and left open a number of questions regarding the 'other' definition. This 'other' definition is the subject of the present paper and we attempt to emulate some of Enochs' work for the category of acts over monoids and concentrate, in the main, on strongly flat acts. We hope to extend this work to other classes of acts, such as injective, torsion free, divisible and free, in a future report.

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“…(The open problem 5 in [9].) Later in [5], the authors characterized monoids over which every right S-act has a strongly flat (condition (P)) cover. They also gave a new characterization for perfect monoids as monoids over which every strongly flat right S-act has a projective cover.…”

Section: Introductionmentioning

confidence: 99%