 Advances in Pure Mathematics, 2012, 2, 104-108 http://dx.doi.org/10.4236/apm.2012.22014 Published Online March 2012 (http://www.SciRP.org/journal/apm) On P-Regularity of Acts Akbar Golchin, Hossein Mohammadzadeh, Parisa Rezaei Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran Email: agdm@math.usb.ac.ir Received September 27, 2011; revised December 17, 2011; accepted December 30, 2011 ABSTRACT 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. Keywords: P-Regularity; Rees Factor Act 1. Introduction Throughout this paper will denote a monoid. We refer the reader to () and () for basic results, definitions and terminology relating to semigroups and acts over monoids and to [3,4] for definitions and results on flat-ness which are used here. SS,A monoid is called left (right) collapsible if for every sszs zsS there exists such that zS szszS,,. A submonoid of a monoid is called weakly left collapsible if for all PssP zS the equality szsz implies that there exists an element such that . uPus,us SA monoid is called right (left) reversible if for every ssS, there exist such that ,uvSus vs .susklK ,\ ,.Sx ySKxtytKxtyt v A right ideal K of a monoid S is called left stabilizing if for every , there exists such that and it is called left annihilating if, Klk kt If for all ,\stSK:SS and all homomorphisms fSs StS   ,fsft Kfsft then is called strongly left annihilating. KA right S-act A satisfies Condition if for Pas a s,,A,aa,ssSS implies the existence of such that and ,,uvaA ,aaua av ,aa A.usvsA right S-act A is called connected if for  there exist 11,,, ,nnststS11,,naaA and 11112 221nn nasa tas atas atSFPF WPFWKF PWKFTKF such that We use the follow ing abbreviations: Strong flatness = ; Pullback flatness = ; Weak pullback flatness = ; Weak kernelflatness = ; Principal weak kernelflatness =; Translation kernelflatness = ; WP ; Weak homoflatness = Principal weak homoflatness = PWPWF PWFTFS; Weak flatness = ; Principal weak flatness =; Torsion freeness = . 2. Characterization by P-Regularity of Right Acts Definition 2.1. Let be a monoid. A right S-act A is called P-regular if all cyclic subacts of A satisfy Condi-tion P. We know that a right S-act A is regular if every cy-clic subact of A is projective. It is obvious that every regular right act is P-regular, but the converse is no t true, for example if is a non trivial group, then is right reversible, and so by ([2, III, 13.7]), S is P-regular, but by ([2, III, 19.4]), S SS is not regular, since has no left zero element. SSSTheorem 2.1. Let be a monoid. Then: 1)  is P-regular if and only if is right reversi- ble. SSS2) S is P-regular if and only if all principal right ideals of satisfy Condition P. A is a right S-act and iAiI, are subacts of 3) If ,A, then iI iA is P-regular if and only if iA is P-regular for every iI. 4) Every subact of a P-regular right S-act is P-regular. Proof. It is straightforward. q.e.d. Here we give a criterion for a right S-act to be P- Copyright © 2012 SciRes. APM A. GOLCHIN ET AL. 105regular. Theorem 2.2. Let be a monoid and SA a right S- act. Then A is P-regular if and only if for every aA and ,,xyS ax ay,uv S implies that there exist  such that and . aauav ux vyProof. Suppose that A is a P-regular right S-act and let , for and ax ay aA,xySaS. Then satis-fies Condition . But PkerSaaS, and so by ([2, III, 13.4]), we are done. Conversely, we have to show that satisfies Con-dition aSP for every a. Since AkeraS Sa, then it suffices to show that ker aS satisfies condition (P) and this is true by ([2, III, 13.4]) . q.e.d. We now give a characterization of monoids for which all right S-acts are P-regular. Theorem 2.3. For any monoid the following state- ments are equivalent: S PPPx0SS0,1) 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. 5) 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)  (3) (4) and (3) (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 adjoin ed. (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. (6) (1). By ([2, IV, 9.9]), all cyclic right S-acts satisfy condition , and so by definition all right S- acts are P-regular as required. q.e.d. Notice that freeness of acts does not imply P-regu-larity, for if , with , then is free, but S is not P-regular, otherwise 0, 1,S2xSxSxS satis-fies Condition as a cyclic subact of S, and so P..0xxx, implies the existence of such that ,uv Sxxu xv and , and this is a contradiction. 0ux vTheorem 2.4. For any monoid the following state- ments are equivalent: S1) All right S-acts satisfying Condition EE are P- regular. 2) All finitely generated right S-acts satisfying Condi-tion 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 P- regular. 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 P- regular. 13) All free right S-acts are P-regular. 14) All finitely generated free right S-acts are P-regu-lar. 15) All free cyclic right S-acts are P-regular. 16) All principal right ideals of S satisfy Condition P. ,,stz S 17)   ,.zsztu vSzzuzvusvt       SP Proof. Implications (1) (2) (3)  (6) (9) (12) (15), (1)  (4) (5)  (6), (4) (7) (8) (9), (7) (10) (11) (12) and (10) (13)  (14) (15) are obvious. (15) (16). As a free cyclic right S-act S is P-regular, and so by (2) of Theorem 2.1, all principal right ideals of S satisfy Condition .  (17). By ([2, III, 13.10]), it is obvious. (16) (17) (1). Suppose the right S-act A satisfies Condition Eax ayand let , for and aA,xyS. Then there exist and such that aAuSaau and ux uy. Thus by assumption there exist stS uusut. and s such that ,xty,aaua usas Therefore  ,aauautat  sxty, and so by Theorem 2.2, A is P-regular. q.e.d . Notice that cofreeness does not imply P-regularity, otherwise every act is P-regular, since by ([2, II, 4.3]), every act can be embedded into a cofree act. But if 0,1,,Sx20,x with  then as we saw before, is not P-regular, and so we have a contradiction. SSS  Theorem 2.5. For any monoid the following state- ments are equivalent: 1) All divisible right S-acts are P-regular. 2) All principally weakly injective right S-acts are P- regular. 3) All fg-weakly injective right S-acts are P-regular. 4) All weakly injective right S-acts are P-regular. 5) All injective right S-acts are P-regular. 6) All injective cogenerators in Act-S are P-regular. 7) All cofree right S-acts are P-regular. 8) All right S-acts are P-regular. 9) S is a group or a group with a zero adjoined. Proof. Implications (1) (2) (3)  (4) (5) (6) and (5) (7) are obvious. (6) (8). Suppose that A is a right S-act, is an injective cogenerator in Act-S and is an injective envelope of A (C exists by [2, III, 1 .23]). By ([5, Theo- rem 2]), BCDBCD is an injective cogenerator in Act-S, and so by assumption is P-regular. Since AC, we have A is P-regular. AD, and so by Theorem 2.1, Copyright © 2012 SciRes. APM A. GOLCHIN ET AL. 106 (7) (8). Let A be a right S-act. Then 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, A is P-regular. (8) (9). By Theorem 2.3, it is obvious. x20x(8) (1). It is obvious. q .e.d. Theorem 2.6. Let S be a monoid. Then every strongly faithful right S-act is P-regular. Proof. By Theorem 2.2, it is obvious. q.e.d. Although strong faithfulness implies P-regularity, but faithfulness does not imply P-regularity, since every mo- noid as an act is faithful, with 0, 1,S is faithful, but as we saw before, is not P-regular. Now see the following theorem. SS.Theorem 2.7. For any monoid S the following state-ments are equivalent: 1) All faithfull right S-acts are P-regular. 2) All finitely generated faithfull right S-acts are P- regular. 3) All faithfull right S-acts generated by at most two elements are P-regular. 4) S is a group or a group with a zero adjoined. Proof. Implications (1) (2)  (3) are obvious. (3) (4). By Theorem 2.3, it suffices to show that every cyclic right S-act is P-regular. Thus we consider a cyclic right S-act and let SSaSAaSSS Since S is faithful, SA is faithful, also SA is generated by at most two elements, thus by assumption SA is P-regular. Since is a subact of aS SA, by (4) of Theorem 2.1, is P-regular as required. aS(4) (1). By Theorem 2.3, it is obvious. q.e.d. SSince regularity does not imply flatness in general, P-regularity also does not imply flatness in general, but as the following theorem shows, for regular monoids P- regularity implies flatness. Theorem 2.8. Let S be a regular monoid. Then every P-regular right S-act is flat. Proof. Suppose that S is a regular monoid, M is a left S-act and SA is a P-regular right S-act. Let in Sam a mAMam for S and We show holds also in S,aaaAm,SM..mmASm Sm Since in ama mSAM, we have a tossing 1122113 32 2 ''kksm msmt msm tmm tm11,,,kk11112 221 kk kas a tas atas at of length , where k,,sstt S.kSM1,k, 11,, ,kSaaA1,,mmIf  then we have 1111 11.smmasa tmt mS1asa t Since is regular, the equality 1at atss implies that 1 111, for 11.Vs Since SsA is P-re- gular, there exist SaA and uv such that ,Saauav 1111.vt s s and ut From the last equality we obtain 1 1111 111.umutm vtssm vtsm11msm Since , we get 11 ,ssmm and so we have 111 11 111 1111''amassmassmat smautsm avtsm avtsmaumaumam    ASm Smin S2k.k1asa t. We now suppose that and that the required equality holds for every tossing of length less than From 11 we obtain equalities 11 1111at atss for 11sVs111asas tt and 1 for 1. Since 1tVtSA is P-regular, there exist 12,SaaA12,,12,vv S and uu  such that 11111 ,aauav ut vtss 1111112 12 111,.aauavus vstt and 222 2 21 1222111112211 1usmumausaut usmutm 122111112223 32 21 kk kkkusm utmausat sm tmasatm tm 1.k Thus we have the fol-lowing tossing of length 1 and of length  From the tossing of length 1, we have 22 1122aumausm S in AM1122aumausm , and so we have 222122S in ASumSusm122111 11111 111,usmutm vtssmvtsmSm. Since 22 1122aumausm we have Sin ASm Sm1k1111autmam. Also from the tossing of length , we have  S in AM. Thus we have 1111autmamA in 11 1SSut mSm11111 1,utmvtsm Sm Since 1111autmamwe have ,S in ASm Sm2222112 21111amaumauma usmautmam and so     SASm Smin  as required. q.e.d. Copyright © 2012 SciRes. APM A. GOLCHIN ET AL. 1073. 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 SK a right ideal of S. Then S is P-regular if and only if SSKKS and S is right reversible or 1KS and all principal right ideals of S satisfy Condition .P Proof. Let SK be a right ideal of S and suppose that SSK is P-regular. Then S satisfies Condition (P) If SSKKS, then by ([2, III, 13.7]), S is right reversible, otherwise by ([2, III, 13.9]), 1KS, and so SSK S. Thus by (2) of Theorem 2.1, all principal right ideals of S satisfy Condition .P Conversely, suppose that SK is a right ideal of If S.SKS and S is right reversible, then by (1) of Theo-rem 2.1, SS is P-regular. If SK 1SK and all principal right ideals of satisfy Condition SP, then by (2) of Theorem 2.1, SSK S is P-regular. q.e.d. Although freeness of acts implies Condition P,x20x in general, but notice that freeness of Rees factor acts does not imply P-regularity, for if with 0, 1,S, and S0,KS then 0SSK SSS as a Rees fac-tor act is free, but as we saw before, is not P-regu- lar. S SSSNow let see the following theorem. Theorem 3.2. Let be a monoid and U be a property of S-acts implied by freeness. Then the follow-ing statements are equivalent: 1) All right Rees factor S-acts satisfying property U -regular. are P2) All right Rees factor S-acts satisfying property U fy Condition P anither S contains no left zero or all principal right ideals of S satisfy Condition satis d eP. Proof. -acts satisfyin(1)  (2). By definiti Rees factor gon all rightS property U satisfy Condition .P Suppose now that S containseft zero 0z. Then 00S a lKzS z is a right ideal of ,S and so SSS. by as-p SK Since). SS is free, SS P-regular,and so alrincipal right ideals of S satisfy Condition P. (2)  (1Let issumtion,l pSSK erty satisfies propU for the rigideal Sht K o Then by assumptionf S.SK satisfies Conditi PNow there are two cas follows: Case 1Son . es as. SKS. Then SSSK so by ([2, III, le, thus by  , and 13.7]), S reversib(1) of Theorem 2.1, is right SSSK  is P-regular. SCase 2. K is a proper right ideal of S. Then by ([2, III, 13.9]), 1S. Thus 0,SKKz fo some 0zSr, and so 0z ption all prl right ides of S satisfy Condition P, that is is left zero. Thus by assuincipaal mSSSK S is P-regula r . q.e. d. y 3.1. For any monoiCorollar d S the following state- mfactor S-acts satisfying Condition ents are equivalent: 1) All right Rees  are P-regular. ) All WPF righP2t Rees factor S-acts are P-regular. PF SFroj lar. P-e right Rees factor S-acts are P-regular. deals of s3) All right Rees factor S-acts are P-regular. 4) All right Rees factor S-acts are P-regular. 5) All pective right Rees factor S-acts are P-regu6) All Rees factor projective generators in Act-S are regular. 7) All fre8) S contains no left zero or all principal right i Satisfy Condition .P P oof. By Theorem 3.2s obr, it ivious. q.e.d. ing state- mees factor S-acts are P-regular. lat rl SCorollary 3.2. For any monoid S the followents are equivalent: 1) All WF right R2) All fight Rees factor S-acts are P-regular. 3) S is not right reversible or no proper right ideaK, 2S of S is left stabilizing, and if S contain , the all principal right ideals f S satisfy Condition Ks aleft zerono.P Proof. Itw follos from Theorem 3.2, ([2, IV, 9.2]), and th. For any monoid S the following state- mRees factor S-acts are P-regular. is righat for Rees factor acts weak flatness and flatness are coinside. q.e.d. Corollary 3.3ents are equivalent: 1) All PWF right 2) S t reversible, no proper right ideal SK, 2Ks left zero aS of S is left stabilizing, and if S containa , thenll princip al right ideals of Ssatisfy Con-dition .P Proo fof. It llows from Theorem 3.2, and ([2, IV, 9.7]). q. llary 3.4. For any monoid S the following state- mes factor S-acts are P-regular. er tive mriSe.d. Coroents are equivalent: 1) All TF right Re2) EithS is a right reversible right cancellaonoid or a ght cancellative monoid with a zero ad-joined, and if S contains a left zero, then all principal right ideals of satisfy Condition .P Proof. It folls from Theorem 3.2, ad own([2, IV, 9.8]). q. ollary 3.5. For any monoid S the following st r S-acts satisfying Condition e.d. Coratements are equivalent: 1) All right Rees factoP are P-regular. S is not right rW2) eversible or no proper right ideal SK, 2S of S is left stabilizing and strongly left ihnd if S contains a left zero, then all prin-cipal right ideals of satisfy Condition Kann ilating, aS.P Proof. It follows m Theorem 3.2, and fro([3, Proposi-Copyright © 2012 SciRes. APM A. GOLCHIN ET AL. Copyright © SciR APM 108 tio o monoid S the following state- mfactor S-acts satisfying Condition P2) S ie and no propeght ideal S2012 es. n 3.26]). q.e.d. Corollary 3.6. Fis (2)  (3)  (4) are ob connected as a left S-act. Proof. Implications (1) r anyents are equivalent: 1) All right Rees vious. (1)  (5). By Theorem 3.3, and ([4, Corollary 24]) it vioWP are P-regular. s right reversiblis obus. (2)  (6). BKr ri, 2S of S is left stabilizing and left ann ihilating,d ains a left zero, then all principal right ideals of S satisfy Condition P. Proof. It followy Theorem 3.3, and ([6, Proposition 8]) it vioK anif S conts from Theorem 3.2 ([3, Corollary 3. nsider monoids over which P-regularity of RU be a prop-er n a, andis obus. (4)  (7). By Theorem 3.3, and ([6, Proposition 7]) it vio. By ([6, Proposition 28]), is obus. (4)  (1)27]). q.e.d. Here we co.PFPTKF Now if AS is a P-regut is obvious that SWlar right Rees factor S-act, then iA satisfies Condi- tion P, also by assumption SA is,KF and so Sees factor acts implies other properties. Theorem 3.3. Let S be a monoid and TA is W q.e.d. Corollary 3.10..PF For any monoid S the following state- mty of S-acts implied by freeness. Thell P-regular right Rees factor S-acts satisfy property U if and only if S is not right reversible or S satisfies operty (U). Proof. Suppose that S is ht reversible. By (1) ofents are equivalent: 1) S is .WPF pr rigTheorem 2.1, SSSSatisfies pr is P-regular, and so by as-sumption S soperty U. Conversel, suppose ySSK is egP-rular for the right ideal SK of S. Then thre two cases as follows: Cas. Sere ae 1KS. Then SSSK  is P-regular, and S is right rso by (1) ofm 2.1, eversible. Thus by assumption TheoreSSSK satisfies property (U). Case 2. SK is a prideal of S. By Theo-oper t righrem 3.1, 1K, and so SSSSK S. Ths uSSK is free, and so satisfies property . q.e.onU men all P-regu- la ac from Teorem 3.3, and ([2, I, 5.23]). q. ollary 3.8. Let S be a monoid. Then all P-regular rig2, III, 17.2 ]). q. ollary 3.9. Let S be a monoid. Then all P-regular rig ([2, III, 14q. orem 3.4. For any monoid S the following state-mt Rees factor S-acts are WPF. keS7) S is not right reveev,zS kerd Corollary 3.7. Let S be aoid. Thr right Rees factor S-ts are free if and only if S is not right reversible or 1S. Proof. It followshe.d. Corht Rees factor S-acts are projective if and only if S is not right reversible or S contains a left zero. Proof. It follows from Theorem 3. 3, and ([e.d. Corht Rees factor S-acts are strongly flat if and only if S is not right reversible or S is left collapsible. Proof. It follows from Theorem 3. 3, and.3]). e.d Theents are equivalent: 1) All P-regular righ2) All P-regular right Rees factor S-acts are WKF. 3) All P-regular right Rees factor S-acts are PWKF 4) All P-regular right Rees factor S-acts are TKF. 5) S is not right reversible or S is weakly left collapsible. 6) S is not right reversible or for every left ideal I of S, rf is connected for every homomorphism :.SfI S rsible or for ery z 2) S is .WK FS ht revble and weakly left collapsible. S3) is rigersi4) is right reversible and for every left ideal I of SrfI5) S is rit reversible and for every ,zS ker, kef is connected for every homomorphism :.SSS ghz is ne (2) is obvious. concted as a left S-act. Proof. Implication (1) (1)  (3). It is obvioby ([6, Corollary us 24]). (3)  (4)  (5). It is obvious by Theorem 3.4. (3)  (4). I o bvious by ([6, Proposition 8]). q.e.dt is. llaThCorory 3.11. Let S be a right reversible monoid. en S is WPF if ad only if S is .TKF ProoIt is ious that every PF rSnf. obvWight -act is T.KF If S is ,TKF then by ([6osition 7]), for zS , Propevery , kerz islary connected as a left S-act, and so by Corol 3.10 S is .WPF q.e.d. REFERENCES  J. M. Howie, “roup Theory,” Cla- Mikhalev, “Monoids, Acts Fundamentals of Semigrendon Press, Oxford, 1995.  M. Kilp, U. Knauer and A.and Categories,” W. de Gruyter, Berlin, 2000. doi:10.1515/9783110812909  V. Laan, “Pullbacks and Flatness Properties of Acts,” perties of Acts I,” Tartu University Press, Tartu, 1999.  V. Laan, “Pullbacks and Flatness ProCommunications in Algebra, Vol. 29, No. 2, 2001, pp. 829- 850. doi:10.1081/AGB-100001547  P. Normak, “Analogies of QF-Ring for Monoids. I,” Tar- ks and tu Ülikooli Toimetised, Vol. 556, 1981, pp. 38-46.  S. Bulman-Fleming, M. Kilp and V. 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