Strongly Torsion Free Acts Over Monoids

Zare, Abas; Golchin, Akbar; Mohammadzadeh, Hossein

doi:10.1142/s1793557113500496

Cited by 6 publications

(5 citation statements)

References 8 publications

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“…(3) ⇒ (1) Every R-torsion free S-act is principally weakly flat, by [21,Theorem 4.5]. Since every regular monoid is left PP and so is left PSF, principally weakly flat is equivalent to Condition (P W P ssc ), by part (2) of Theorem 2.8.…”

Section: Recall That For Any Nonempty Set I S Imentioning

confidence: 86%

“…Also every left almost regular monoid is left PP, by the duality of [14, IV, Proposition 1.3], and so is left PSF. Thus all torsion free S-acts satisfy Condition (P W P ssc ), by part (2) of Theorem 2.8.Recall, from[21], that A is called R-torsion free if for any a, b ∈ A and c ∈ S, c right cancellable, ac = bc and a R b (R is Green's equivalence) imply that a = b.…”

mentioning

confidence: 93%

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Classification of monoids by Condition $(PWP_{ssc})$

Khamechi

Baluchestan

Saany

et al. 2020

CGASA

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Condition (P W P) which was introduced by Laan is related to the concept of flatness of acts over monoids. Golchin and Mohammadzadeh introduced Condition (P W PE) as a generalization of Condition (P W P). In this paper, we introduce Condition (P W Pssc) which is much easier to check than Conditions (P W P) and (P W PE) and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition (P W Pssc) will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition (P W Pssc) and also a classification of monoids S will be given for which all right S-acts satisfying some other flatness properties have Condition (P W Pssc).

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Section: Recall That For Any Nonempty Set I S Imentioning

confidence: 86%

mentioning

confidence: 93%

Classification of monoids by Condition $(PWP_{ssc})$

Khamechi

Baluchestan

Saany

et al. 2020

CGASA

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“…Te notions of torsion free and divisible S-acts are known and defned by using the right and left cancellable elements of S, respectively (see [16]). In [2], torsion freeness and divisibility are considered in a much stronger sense (without imposing the cancellability properties on elements of S ) which we call here strong torsion freeness (see also [20]) and strong divisibility defned as follows.…”

Section: □ Theorem 15 Let S Be a Left Reversible Monoid Ten (I) Any D...mentioning

confidence: 99%

Characterizing Topologically Dense Injective Acts and Their Monoid Connections

Hezarjaribi Dastaki,

Rasouli,

Barzegar

2024

Journal of Mathematics

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In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free S-act over a monoid S is topologically dense injective if and only if S is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided.

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“…We recall from [13] that a right S-act A S is strongly torsion free if the equality as = bs, for all a, b ∈ A S and all s ∈ S, implies a = b. Theorem 3.10. For any monoid S the following statements are equivalent:…”

Section: Characterization Of Monoids By U -(G-p W P ) Of Right Actsmentioning

confidence: 99%

On the property U-(G-PWP) of acts

Arabtash

Golchin

Mohammadzadeh

2018

CGASA

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In this paper first of all we introduce Property U-(G-P W P) of acts, which is an extension of Condition (G-P W P) and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong faithfulness (Pcyclicity) and (P-)regularity of acts imply the property U-(G-P W P). Finally, we give a necessary and sufficient condition under which all cyclic (finitely generated) right acts or all (strongly,-) torsion free cyclic (finitely generated) right acts satisfy Property U-(G-P W P).

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Strongly Torsion Free Acts Over Monoids

Cited by 6 publications

References 8 publications

Classification of monoids by Condition $(PWP_{ssc})$

Classification of monoids by Condition $(PWP_{ssc})$

Characterizing Topologically Dense Injective Acts and Their Monoid Connections

On the property U-(G-PWP) of acts

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