On a generalization of condition (E)

Abstract

Golchin and Renshaw in [A flatness property of acts over monoid, Conf. Semigroup, University of St. Andrews, 1998, pp. 72–77] introduced Condition [Formula: see text] which is a generalization of Condition (P) and showed that Condition [Formula: see text] implies weak flatness, but not the converse. In [A. Golchin and H. Mohammadzadeh, On homological classification of monoids by Condition [Formula: see text] 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 [Formula: see text], 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 [Formula: see text] coincides with Condition (P) (respectively, pullback flatness, strong flatness) of acts. Furthermore, many known results are generalized.