Asian-European Journal of Mathematics Vol. 16, No. 2 (2023) 2350021 (17 pages) c  World Scientific Publishing Company DOI: 10.1142/S1793557123500213 On a generalization of condition (E) Zahra Jafari Maskoni ∗ , Akbar Golchin † and Hossein Mohammadzadeh Saany ‡ Department of Mathematics University of Sistan and Baluchestan Zahedan, Iran ∗ zahrajafari65@yahoo.com † agdm@hamoon.usb.ac.ir ‡ hmsdm@math.usb.ac.ir Communicated by R. P. Sharma Received July 26, 2018 Revised February 15, 2021 Accepted February 16, 2021 Published 26 July 2022 Golchin and Renshaw in [A flatness property of acts over monoid, Conf. Semigroup, University of St. Andrews, 1998, pp. 72–77] introduced Condition (P E ) which is a gen- eralization of Condition (P ) and showed that Condition (P E ) implies weak flatness, but not the converse. In [A. Golchin and H. Mohammadzadeh, On homological clas- sification of monoids by Condition (P E ) of right acts, Italian J. Pure Appl. Math. 25 (2009) 175–186] characterized monoids by this condition of their acts. In this paper, we first introduce Condition (E E ), which is a generalization of Condition (E) and then give a characterization of monoids by this condition of their (cyclic, Rees factor) acts. Also, we give a necessary and sufficient condition under which Condition (E E ) coincides with Condition (P ) (respectively, pullback flatness, strong flatness) of acts. Furthermore, many known results are generalized. Keywords : Condition (E E ); Rees factor acts; E E -left annihilating. AMS Subject Classification: 20M30, 20M50 1. General Appearance Laan in [7] introduced Condition (E  ), a generalization of Condition (E). Golchin and Renshaw [3] introduced a generalization of Condition (P ) called Condition (P E ). After that Golchin and Mohammadzadeh [2] provided a characterization of monoids by this condition of their right acts. In this paper, we introduce a ∗ Corresponding author. 2350021-1