On $GPW$-Flat Acts

Document Type: Research Paper

Authors

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.

Keywords

References

[1] Bulman-Fleming, S., Kilp, M., and Laan, V., Pullbacks and flatness properties of acts II, Comm. Algebra 29(2) (2001), 851-878.
[2] Golchin, A., Flatness and coproducts, Semigroup Forum 72(3) (2006), 433-440.
[3] Golchin, A., On flatness of acts, Semigroup Forum 67(2) (2003), 262-270.
[4] Golchin, A. and Mohammadzadeh, H., On Condition (P0), Semigroup Forum 86(2) (2013), 413-430.
[5] Golchin, A. and Mohammadzadeh, H., On regularity of Acts, J. Sci. Islam. Repub. Iran 19(4) (2008), 339-345.
[6] Golchin, A., Zare, A., and Mohammadzadeh, H., E-torsion free acts over monoids, Thai J. Math. 14(1) (2015), 93-114.
[7] Kilp, M., On flat acts (Russian), Tatru UL. Toimetisted, 253 (1970), 66-72.
[8] Kilp, M., Characterization of monoids by properties of their left Rees factors, Tatru UL. Toimetisted, 640 (1983), 29-37.
[9] Kilp, M., Knauer, U., and Mikhalev, A., “Monoids, Acts and Categories”, De Gruyter, 2000.
[10] Laan, V., Pullbacks and flatness properties of acts I., Comm. Algebra 29(2) (2001), 829-850.
[11] Nouri, L., Golchin, A., and Mohammadzadeh, H., On properties of product acts over monoids, Comm. Algebra 43(5) (2015), 1854-1876.
[12] Qiao, H., Some new characterizations of right cancellative monoids by Condition (PWP), Semigroup Forum 71(1) (2005), 134-139.
[13] Qiao, H., Limin, W., and Zhongkui, L., On some new characterizations of right cancellative monoids by flatness properties, Arab. J. Sci. Eng. 32(1) (2007), 75-82.
[14] Sedaghatjoo, M., Khosravi, R., and Ershad, M., Principally weakly and weakly coherent monoids, Comm. Algebra 37(12) (2009), 4281-4295.
[15] Zare, A., Golchin, A., and Mohammadzadeh, H., Strongly torsion free acts over monoids, Asian-Eur. J. Math. 6(3) (2013), 1350049.