Thai Journal of Mathematics Volume 14 (2016) Number 1 : 93–114 http://thaijmath.in.cmu.ac.th ISSN 1686-0209 E -Torsion Free Acts Over Monoids Akbar Golchin, Abbas Zare 1 and Hossein Mohammadzadeh Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran e-mail : agdm@math.usb.ac.ir (A. Golchin) e-mail : zareh.abbas@gmail.com (A. Zare) e-mail : hmsdm@math.usb.ac.ir (H. Mohammadzadeh) Abstract : In this paper we introduce E-torsion freeness of acts over monoids, and will give a characterization of monoids by this property of their (cyclic, mono- cyclic, Rees factor) acts. Keywords : S-act; E-torsion freeness; flatness. 2010 Mathematics Subject Classification : 20M30. 1 Introduction Throughout this paper S will denote a monoid. We refer the reader to [1] and [2] for basic definitions and terminology relating to semigroups and acts over monoids and to [3], [4], [5] and [6] for definitions and results on flatness which are used here. A monoid S is called left(right) collapsible if for any s, s ′ ∈ S there exists z ∈ S such that zs = zs ′ (sz = s ′ z). A submonoid P of S is called weakly left collapsible if for any s, s ′ ∈ P , z ∈ S, sz = s ′ z implies the existence of u ∈ P such that us = us ′ . It is obvious that every left collapsible submonoid is weakly left collapsible, but the converse is not true. A monoid S is called right (left) reversible, if for any s, s ′ ∈ S, there exist u, v ∈ S such that us = vs ′ (su = s ′ v). A submonoid P of S is called weakly right reversible, if for any s, s ′ ∈ P , z ∈ S, sz = s ′ z implies the existence of u, v ∈ P such that us = vs ′ . A right ideal K S of a monoid S is called left stabilizing, if for any k ∈ K S , there exists l ∈ K S such that lk = k. K S is called left annihilating, if for any t ∈ S, x, y ∈ S \ K S , xt, yt ∈ K S 1 Corresponding author. Copyright c  2016 by the Mathematical Association of Thailand. All rights reserved.