On the U-WPF Acts over Monoids

Transcription

1 Journal of cence, Ilamc Republc of Iran 8(4): (007) Unverty of Tehran, IN On the U-WPF ct over Monod. Golchn * and H. Mohammadzadeh Department of Mathematc, Unverty of tan aluchetan, Zahedan, Ilamc Republc of Iran btract Vald Laan n [5] ntroduced an extenon of trong flatne whch called wea pullbac flatne. In th paper we ntroduce a new property of act over monod, called U-WPF whch an extenon of wea pullbac flatne and gve a clafcaton of monod by th property of ther act and alo a clafcaton of monod when th property of act mple other. We alo how that regularty and trong fathfulne of act both mply U-WPF. n equvalent of that over monod for whch toron freene mple U-WPF gven too. Keyword: U-WPF; trongly fathful; Regular. Introducton and Prelmnare We deal n th paper wth what generally referred to a homologcal clafcaton of monod by flatne properte. Many paper have appeared recently nvetgatng the condton on a monod whch are neceary and uffcent to mae a varety of the flatne properte concde, ether for all rght act, or for all rght act of certan type. Condton (P) and ( ) of act over monod have appeared n many paper and monod are clafed when thee properte of act mply other flatne properte and vce vera. rght act trongly flat f t atfe condton (P) and ( ). Vald Laan n [5] ntroduced an extenon of trong flatne whch called wea pullbac flatne. In th paper we ntroduce a new property of act over monod, called U-WPF whch an extenon of wea pullbac flatne and gve a clafcaton of monod by th property of ther act and alo a clafcaton of monod when th property of act mple other. We alo how that regularty and trong fathfulne of act both mply U-WPF. n equvalent of that over monod for whch toron freene mple U-WPF gven too. Throughout th paper wll denote a monod. We refer the reader to [3], [4], for bac defnton and termnology relatng to emgroup and act over monod and to [6], [7] for defnton and reult on flatne whch are ued here. monod left collapble f for every,t there ext u uch that u ut. monod atfe Condton (K) f every left collapble ubmonod of contan a left zero. monod rght PP f every prncpal rght deal of projectve. If for every there ext x uch that x, then called a regular monod, for example for any feld K and any n N the monod (Mat ( K ),.) of all n n matrce wth entre n K n a regular monod. rght deal K of left tablzng f for every K there ext l K uch that l. If I a proper rght deal of, then B I.. {( ax, ) a \ I} I {( ay, ) a \ I} * Correpondng author, Tel.: +98(54)44766, Fax: +98(54)44766, -mal: agdm@math.ub.ac.r 33

2 Vol. 8 No. 4 utumn 007 Golchn and Mohammadzadeh J. c. I. R. Iran wth ( a, z ), ( az, ) a a I otherwe for every a \ I, and z { x, y} a rght - act. (Note that by. we mean djont unon.) rght -act atfe Condton (P) f for all aa,,,, a a mple that there ext a, u, v uch that a au, a av and u v. It atfe Condton ( ) f for all a,,, a a mple that there ext a, u uch that a a u and u u. If for all a,,, z, a a and z z mply that there ext a, u uch that a a u and u u, then t atfe Condton ( ). rght -act wealy pullbac flat f t atfe both Condton (P) and ( ). rght -act atfe Condton ( P ) f whenever a, a,,, and a a there ext a and u, v, e e, f f uch that ae a ue, af a vf, e, f and u v. It hown n [] that Condton ( P ) mple wea flatne, but the convere not true. rght -act trongly fathful f for, t the equalty a at for ome a mple that t. If a left cancellatve monod, then a a rght - act trongly fathful. We ue the followng abbrevaton; trong flatne F, wea pullbac flatne WPF, wea ernel flatne WKF, prncpal wea ernel flatne PWKF, tranlaton ernel flatne TKF, wea homoflatne WP, prncpal wea homoflatne PWP, wea flatne WF, prncpal wea flatne PWF.. General Properte Defnton.. Let be a monod. rght -act U-WPF f there ext a famly { B I} of ubact of uch that I B, I WPF. n example of a U-WPF rght -act gven n ecton 3 before theorem 3.. Theorem.. Let be a monod. Then () very WPF rght -act U-WPF. () rght -act U-WPF f and only f for every a there ext a ubact B of uch that a B WPF. (3) If { B I} a famly of ubact of a rght -act uch that for every I B U-WPF. (4) For every proper rght deal I of I, B U-WPF, then I B U-WPF. (5) cyclc rght -act WPF f and only f t U- WPF. Proof. The proof of (), (), (3), (5) traghtforward. Now let I be a proper rght deal of and let I B. Then. C {( a, x ) a \ I} I. {( a, y ) a \ I} I D nce wealy pullbac flat, then C and D are wealy pullbac flat and o B C D U- WPF, thu we have (4). Note that Condton (P) doe not mply U-WPF, otherwe by Theorem., for cyclc rght -act, WPF and Condton (P) concde, whch by ([6, xample 6]), not true n general. 3. Clafcaton of Monod by U-WPF of Rght ct lthough wea pullbac flatne mple toron freene, but note that U-WPF doe not mply toron freene n general, for f (N,.), where N the et of natural number, and f N N N, then by Theorem., U-WPF, but not toron free, otherwe (, x) (, y) mple that (, x) (, y), whch a contradcton. Now t obvou that U- wea pullbac flatne doe not mply wea pullbac flatne and o t natural to a for monod whch U- wea pullbac flatne of ther act mple other properte uch a toron freene. Theorem 3.. For any monod the followng tatement are equvalent: () ll rght -act are toron free. () ll U-WPF rght -act are toron free. (3) ll rght cancellable element of are rght nvertble. Proof. () (). It obvou. () (3). uppoe that c a rght cancelable element of. If c, then obvouly c rght nvertble. Thu we uppoe that c and that 34

3 On the U-WPF ct over Monod c. Then by Theorem., U-WPF. Thu by the aumpton toron free and o c (, x ) c (, y ) c mple that (, x ) (, y ), whch a contradcton. Hence for every rght cancelable element c of,c that, every rght cancelable element of rght nvertble a requred. (3) (). By ([4, IV, 6.]), It obvou. Theorem 3. how, to ee that all act are toron free, t uffce to how that all U-WPF act are toron free. Theorem 3.. For any monod the followng tatement are equvalent: () ll U-WPF rght -act are flat. () ll U-WPF rght -act atfy Condton ( P ). (3) ll U-WPF rght -act are WF. (4) ll U-WPF rght -act are PWF. (5) regular. Proof. Implcaton () (3) (4) are obvou. () (3). nce by ([, Theorem.3]), Condton ( P ) mple wea flatne, then t obvou. (4) (5). Let. If, then t obvou that regular. Thu we uppoe that, and that. By Theorem., U-WPF and o by the aumpton prncpally wealy flat. Thu by ([4, III,.9]), left tablzng and o there ext l uch that l. Hence there ext x uch that l x and o l x that, regular. (5) (). uppoe a U-WPF rght -act. Then there ext a famly { B I} of ubact of uch that B wealy pullbac flat and I B. We how that flat. Thu we uppoe that for the left - act, M for a, a and m, m M. We how that the ame equalty hold alo n ( m m ). nce a m a m n M, then we have a tong m m a at m tm a at 3m 3 t m... a- a t... m t m. of length, where,...,, t,... t, a,..., a -, m,..., m M. If then we have m m a a t m t m. nce regular the equalty a a t mple that at at for V ( ). nce I B, then there ext 0 I uch that, a B. nce B 0 0 WPF, then B 0 atfe Condton (P) and o there ext a and u, v uch that a a u a v and ut vt From the lat equalty we obtan um ut m vt m vt m. nce m m, then m mand o we get a m a m a m at m aut m a utm a vtm a vt m a um a u m a m n ( m m ). uppoe that and that the requred equalty hold for tong of length le than. From a at we obtan equalte at at for V ( ) and a a t t for t V ( t ). nce I B, then there ext, I uch that a B and a B. nce B are WPF, then B atfy Condton (P) and o there ext a B, a B and u, u, v, v uch that a a u a u, ut ut and a a v a v, v v t t. Therefore, we have the followng tong av aut of length and v m v m u m utm u m utm au at 3m 3 t m a- a t m t m. 35

4 Vol. 8 No. 4 utumn 007 Golchn and Mohammadzadeh J. c. I. R. Iran of length. From the tong of length, we have a v m a u m n M. By nductve hypothe we have a v m a u m n ( v m u m ). nce u m utm utm utm m, then we have a v m a u m n ( m m ). lo from tong of the length, we have a utm a m n M. By nductve hypothe we have a m n ( utm m ). nce utm u t m m, then a utm a m n ( m m ). Thu we have a m a v m a v m a u m a utm a m n ( m m ). () (). nce flatne mple prncpal wea flatne, then by the aumpton U-WPF mple prncpal wea flatne and o by the proof of (4) (5), regular. Thu left PP and o by ([, Theorem.5]), wea flatne and Condton ( P ) concde and o we are done. Theorem 3.3. For any monod the followng tatement are equvalent: () ll U-WPF rght -act are WPF. () ll U-WPF rght -act atfy Condton (P). (3) ll U-WPF rght -act are WKF. (4) ll U-WPF rght -act are PWKF. (5) ll U-WPF rght -act are TKF. (6) ll U-WPF rght -act are (WP). (7) ll U-WPF rght -act are (PWP). (8) a group. Proof. Implcaton () (3) (4) (5) (7) and () () (6) (7) are obvou. (7) (8). uppoe that for, and that. Then by Theorem., a U- WPF rght -act and o by the aumpton (PWP). Thu (, x ) (, y ) mple that there ext a and u, v uch that (, x ) au, (, y) av and u v. qualty (, x ) au mple the extence of t \ uch that a ( t, x ). mlarly, a ( t, y ) for ome t \ and o we have a contradcton. Hence for every, and o a group. (8) (). nce a group, then by ([, Propoton 9]), every rght -act wealy pullbac flat and o every U-WPF wealy pullbac flat. Corollary 3.4. For any monod the followng tatement are equvalent: () ll U-WPF rght -act are free. () ll U-WPF rght -act are projectve generator. (3) ll U-WPF rght -act are projectve. (4) ll U-WPF rght -act are trongly flat. (5) {}. Proof. Implcaton () () (3) (4) are obvou. (4) (5). nce every U-WPF rght -act trongly flat, then every U-WPF rght -act WPF and o by Theorem 3.3, a group. Thu by ([, Propoton 9]), every rght -act WPF and o by Theorem., every rght -act U-WPF. Thu by the aumpton every rght -act trongly flat and o by ([4, IV, 0.5]), {}. (5) (). nce {}, then all rght -act are free and o all U-WPF rght -act are free a requred. Note that U-WPF doe not mply trong fathfulne, nce every monod a a rght -act U-WPF, but not trongly fathful, otherwe left cancellatve whch not true n general. Now ee the followng theorem. Theorem 3.5. For any monod the followng tatement are equvalent: () ll U-WPF rght -act are trongly fathful. () ll U-WPF rght -act atfy Condton ( ) and left cancellatve. Proof. () (). nce aumpton U-WPF, then by the trongly fathful and o left cancellatve. lo t obvou that every trongly fathful rght -act atfe Condton ( ) and o by the aumpton every U-WPF rght -act atfe Condton (). () (). uppoe that a U-WPF rght -act and a at for a and, t. nce by the aumpton atfe Condton ( ), then there ext a and u uch that a a u and u ut. But left cancellatve and o t, hence trongly fathful. Note that the above theorem alo true, when rght act n general replaced by fntely generated or cyclc rght act. Lemma 3.6. Let be a monod and a rght -act. If regular, then for all a and, t, a at mple that there ext u uch that a au and u ut. Proof. uppoe that a regular rght -act. Then by 36

5 On the U-WPF ct over Monod ([4, III, 9.]), for every a there ext e ( ) uch that er λa er λe and o a e. By ([4, III, 7.8]), e projectve and o e atfe Condton ( ). Thu a atfe Condton ( ) and o a at mple that there ext w, w uch that a ( aw ) w and w w t. Thu a a( w w ) and ( ww ) ( ww ) t. If ww u, then a au and u ut a requred. Now from Lemma 3.6, we have Corollary 3.7. Let be a monod and a rght -act. If regular, then atfe Condton (). Lemma 3.8. Let be a rght PP monod and a rght -act. If atfe Condton (), then for every a, a atfe Condton (). Proof. uppoe that atfe Condton ( ) and let a at for a,, t. Then there ext a, u uch that a a u, u ut. nce rght PP, then there ext e ( ) uch that er λe er λu. Thu e et, u ue and o a ae. Hence a atfe Condton ( ) a requred. Note that U-WPF doe not mply regularty, for f { 0,, x} wth x 0, then not rght PP and o by ([4, III, 9.3]), and ([4, IV,.5]), not regular, but t obvou that a U-WPF cyclc rght act. Now t natural to a for monod over whch every U-WPF rght -act regular. Theorem 3.9. Let be a monod. Then every U-WPF rght -act regular f and only f rght PP, t atfe Condton (K) and every WPF rght -act trongly flat. Proof. nce U-WPF, then by the aumpton regular and o by ([4, III, 9.3]), all prncpal rght deal of are projectve. Thu by ([4, IV,.5]), rght PP. lo by the aumpton all trongly flat cyclc rght -act are regular. Thu by ([4, III, 9.3]), all trongly flat cyclc rght -act are projectve and o by ([4, IV,.]), atfe Condton ( K ). nce by Corollary 3.7, every regular rght -act atfe Condton (), then by the aumpton and Theorem., every WPF rght -act atfe Condton () and o every WPF rght -act trongly flat. Converely, uppoe that a U-WPF rght -act. Then there ext a famly { B I} of ubact of uch that IB, I WPF. Thu by the aumpton every B F and o atfe Condton ( ). Now let a. Then there ext 0 I uch that a B. nce rght PP 0 0 atfe condton ( ), then by Lemma 3.8, a atfe Condton ( ) and o a F. nce atfe Condton ( K ), then by ([4, IV,.]), a projectve and o by ([4, III, 9.3]), regular a requred. Note that f ( N,.), where N the et of natural number, then N left cancellatve. It can ealy be een that N N N trongly fathful. nce the rght deal N of N not left tablzng, then by ([4, III,.9]), not PWF and o not WPF. Hence trong fathfulne doe not mply wea pullbac flatne n general, but by the followng theorem t can be een that trong fathfulne mple U-WPF. Theorem 3.0. Let be a monod and a rght -act. If trongly fathful, then U-WPF. Proof. uppoe that a and let ψ a : a be uch that a. nce trongly fathful, then ψ a well-defned and o t an omorphm that, a for every a. Thu all cyclc ubact of are WPF. nce a, then U-WPF. a Theorem 3.. Let be a monod and a rght -act. If regular, then U-WPF. Proof. uppoe that a regular rght -act. Then by ([4, III, 9.3]), all cyclc ubact of are projectve and o all cyclc ubact of are WPF. nce a a, then U-WPF a requred. By the argument after Theorem., we aw that Condton (P) doe not mply U-WPF, thu toron freene doe not mply U-WPF ether. Now we how that monod for whch all toron free rght act are U- WPF are the ame a thoe for whch all toron free cyclc rght act are WPF. Theorem 3.. For any monod the followng tatement are equvalent: () ll toron free rght -act are U-WPF. () ll toron free fntely generated rght -act are U-WPF. (3) ll toron free cyclc rght -act are WPF. Proof. () (). It obvou. () (3). nce by Theorem., for cyclc act WPF and U-WPF concde, then we are done. 37

6 Vol. 8 No. 4 utumn 007 Golchn and Mohammadzadeh J. c. I. R. Iran (3) (). nce all ubact of every toron free rght act toron free, then for a toron free rght -act, a, a alo toron free. Thu by the aumpton for every a, a WPF. Now nce a a, then U-WPF a requred. Now we gve a clafcaton of monod by U-WPF of rght act. Theorem 3.3. For any monod the followng tatement are equvalent: () ll rght -act are U-WPF. () ll fntely generated rght -act are U-WPF. (3) ll cyclc rght -act are U-WPF. (4) a group or a group wth a zero adjoned. Proof. Implcaton () () (3) are obvou. (3) (4). By Theorem., and ([, Propoton 5]), t obvou. (4) (). Let be a rght -act. nce by ([, Propoton 5]), all cyclc rght -act are WPF, then for every a, a WPF. nce a a, then U-WPF a requred. Reference. Bulman-Flemng., Klp M., and Laan V. Pullbac and flatne properte of act II. Comm. lgebra, 9(): (00).. Golchn. and Renhaw J. flatne property of act over monod. Conference on emgroup, Unverty of t. ndrew, July 997, 7-77 (998). 3. Howe J.M. Fundamental of emgroup Theory. London Mathematcal ocety Monograph, OUP (995). 4. Klp M., Knauer U., and Mhalev. Monod, ct and Categore: Wth pplcaton to Wreath Product and Graph: Handboo for tudent and Reearcher. Walter de Gruyter, Berln (000). 5. Laan V. On a generalzaton of trong flatne. cta Comment. Unv. Tartuen, : (998). 6. Laan V. Pullbac and flatne properte of act. Ph.D. The, Tartu. (999). 7. Laan V. Pullbac and flatne properte of act I. Comm. lgebra, 9(): (00). 38