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An act A S is called torsion free if for any a, b ∈ A S and for any right cancellable element c ∈ S the equality ac = bc implies a = b. In [M. Satyanarayana, Quasi-and weakly-injective S-system, Math. Nachr. 71 (1976) 183-190], torsion freeness is considered in a much stronger sense which we call in this paper strong torsion freeness and will characterize monoids by this property of their (cyclic, monocyclic, Rees factor) acts.

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By a regular act we mean an act that all its cyclic subacts are projective. In this paper we introduce P-regularity of acts over monoids and will give a characterization of monoids by this property of their right (Rees factor) acts.