Rashidi, H., Golchin, A., Mohammadzadeh Saany, H. (2019). On $GPW$-Flat Acts. Categories and General Algebraic Structures with Applications, (), -.

Hamideh Rashidi; Akbar Golchin; Hossein Mohammadzadeh Saany. "On $GPW$-Flat Acts". Categories and General Algebraic Structures with Applications, , , 2019, -.

Rashidi, H., Golchin, A., Mohammadzadeh Saany, H. (2019). 'On $GPW$-Flat Acts', Categories and General Algebraic Structures with Applications, (), pp. -.

Rashidi, H., Golchin, A., Mohammadzadeh Saany, H. On $GPW$-Flat Acts. Categories and General Algebraic Structures with Applications, 2019; (): -.

^{1}Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.

^{2}University of Sistan and Baluchestan

^{3}Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Abstract

In this article, we present $GPW$-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right $S$-act $A_{S}$ is $GPW$-flat if for every $s \in S$, there exists a natural number $n = n_ {(s, A_{S})} \in \mathbb{N}$ such that the functor $A_{S} \otimes {}_{S}- $ preserves the embedding of the principal left ideal ${}_{S}(Ss^n)$ into ${}_{S}S$. We show that a right $S$-act $A_{S}$ is $GPW$-flat if and only if for every $s \in S$ there exists a natural number $n = n_{(s, A_{S})} \in \mathbb{N}$ such that the corresponding $\varphi$ is surjective for the pullback diagram $P(Ss^n, Ss^n, \iota, \iota, S)$, where $\iota : {}_{S}(Ss^n) \rightarrow {}_{S}S$ is a monomorphism of left $S$-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.

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