On P-Regularity of Acts

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.


Introduction
Throughout this paper will denote a monoid.We refer the reader to ( [1]) and ( [2]) for basic results, definitions and terminology relating to semigroups and acts over monoids and to [3,4] for definitions and results on flatness which are used here.

Characterization by P-Regularity of Right Acts
1) All right S-acts are P-regular.
2) All finitely generated right S-acts are P-regular.
3) All cyclic right S-acts are P-regular.4) All monocyclic right S-acts are P-regular.
All right Rees factor S-acts are P-regular.6) S is a group or a group with a zero adjoined.Proof.Implications (1) (2 (5) are obvious.(4) (6).By assumption all monocyclic right S- acts satisfy Condition , and so by ([2, IV, 9.9]), S is a group or a group with a zero adjoined.
(5) (6).By assumption all right Rees factor S-acts satisfy Condition and again by ([2, IV, 9.9]), S is a group or a group with a zero adjoined.
Notice that freeness of acts does not imply 2) All finitely generated right S-acts satisfying Condition are P-regular.
3) All cyclic right S-acts satisfying Condition   E are P-regular.
4) All SF right S-acts are P-regular.5) All SF finitely generated right S-acts are P-regular.6) All SF cyclic right S-acts are P-regular.7) All projective right S-acts are P-regular.8) All finitely generated projective right S-acts are Pregular.9) All projective cyclic right S-acts are P-regular.10) All projective generators in Act-S are P-regular.11) All finitely generated projective generators in Act-S are P-regular.
12) All cyclic projective generators in Act-S are Pregular.
13) All free right S-acts are P-regular.14) All finitely generated free right S-acts are P-regular.
15) All free cyclic right S-acts are P-regular.16) All principal right ideals of S satisfy Condition A is P-regular, there exist From the last equality we obtain , we get 1 1 , s s m m   and so we have ' ' a m a s s m as s m a t s m a ut s m a vt s m a vt s m a um a u m a m . We now suppose that and that the required equality holds for every tossing of length less than From 11


we obtain equalities , v v S and u u  such that

k
Thus we have the following tossing of length 1 and of length  From the tossing of length 1, we have a u m a u s m , and so we have . Also from the tossing of length , we have . Thus we have Sm Sm in   as required.q.e.d.

Characterization by P-Regularity of Right Rees Factor Acts
In this section we give a characterization of monoids by P-regularity of right Rees factor acts. Theorem 3.1.Let S be a monoid and S K a right ideal of S. Then S is P-regular if and only if and S is right reversible or ([2, by([2, II,  4.3]), A can be embedded into a cofree right S-act.Since A is a subact of a cofree right S-act, by assumption A is a subact of a P-regular right S-act, and so by Theorem 2.1, 1) All divisible right S-acts are P-regular.2)Allprincipallyweaklyinjectiveright S-acts are Pregular.3)Allfg-weaklyinjectiveright S-acts are P-regular.4)Allweaklyinjective right S-acts are P-regular.5)Allinjectiveright S-acts are P-regular.6)Allinjectivecogenerators in Act-S are P-regular.7)Allcofreeright S-acts are P-regular.Abea right S-act.S S   .Theorem 2.7.For any monoid S the following statements are equivalent: 1) All faithfull right S-acts are P-regular.2)Allfinitelygenerated faithfull right S-acts are Pregular.3)Allfaithfull right S-acts generated by at most two elements are P-regular.4)S is a group or a group with a zero adjoined.
S is not right reversible or no proper right idea K , 2 S  of S is left stabilizing, and if S contain , the all principal right ideals f S satisfy Condition It w follo s from Theorem 3.2, ([2, IV, 9.2]), and th .For any monoid S the following statem Rees factor S-acts are P-regular.is righ at for Rees factor acts weak flatness and flatness are coinside.q.e.d.It llows from Theorem 3.2, and ([2, IV, 9.7]).All TF right Re 2) Eith S is a right reversible right cancella onoid or a ght cancellative monoid with a zero adjoined, and if S contains a left zero, then all principal right ideals of satisfy Condition  . S  of S is left stabilizing and strongly left ih nd if S contains a left zero, then all principal right ideals of satisfy Condition S Case 2. K is a proper right ideal of S .Then by ([P Proof.S  of S is left stabilizing, and if S contain a , then ll principal right ideals of S satisfy Condition  .P Proo fo f.P are P-regular.S is not right r W 2) eversible or no proper right ideal S K , 2 K ann ilating, a S  .P Proof.It follows m Theorem 3.2, and fro ([3, Proposi-