On Annihilator-SPQ-injective module

Transcription

1 IJEIT// Vol03 Iue 07//July//Page o:46-465//i x 205 On Annihilator-PQ-injective module Author Wadbudhe ahatma Fule Art, Commerce itaramji Chaudhari cience ahavidyalaya, Warud Amravati, GB Univerity Amravati [] India - drrameh_wadbude@redifmailcom ABTACT Let be a module with = End (), a right -module i called a-pq-injective if for every - homomorphim from a-mall principal ubmodule of to, can be extended to an -homomorphim from to In thi paper we give characterization propertie of a-pq-injective module ome known reult are extended with annihilator condition Key Word Annihilator mall, mall injective, P-injective, P--injective a-pq-injective Coretractable module ITODUCTIO Let be a ring An -module i called principally injective [ickolon], if for every - homomorphim from principal right ideal of to, can be extended to an -homomorphim from to Equivalently l r (a) = a for all a [Thuyed TC Quynh] introduced the definition of P-injective module, a right - module i called mall principally-injective (briefly, P-injective) if, for every - homomorphim from mall principal ubmodule m to, can be extended to an - homomorphim from to A ring i called right P-injective if i P-injective An - module i called mall principally--injective (briefly, P--injective) if for every - homomorphim from mall principal ubmodule of to, can be extended to an - homomorphim from to If i P-injective, i called PQ-injective [ickolon zhou] introduce annihilator-mall right (left) ideal [TA Kalati DKekin] extend thi notion with ome reult Let be a module K We ay that K i an annihilator-mall ubmodule (a-mall ubmodule) of if K + X =, X i a ubmodule of, implie l (X) = 0, denoted by K a Clearly every mall ub module i annihilator-mall ubmodule but convere i true if i Coretractable module [8] An -module i called Coretractable if, for any proper ub module K of, there exit a non zero homomorphim f : uch that f(k) 0, that i Hom(/K, ) 0 All ring are aociative with identity all module are unitary Let be an -module We Wright, l (m) {m : mr 0, r } r (m) {r : mr 0, m } for each X, the fined by right (left) annihilator of x in i defined by r (X) {r : xr 0, x X} ( l (X) {r : xr 0, x X} ) Foe a ingleton {x} we may Wright r (X) l (X) By notation, e, K, K a, we mean that K i direct umm, an eential, mall annihilator-mall ubmodule of, repectively It i clear that k a if only if k J() Wadbudhe IJEIT Volume 3 Iue 7 July 205 Page 46

2 IJEIT// Vol03 Iue 07//July//Page o:46-465//i x a-pq-ijective Definition2 Let, be an -module A module i called annihilator-mall Principal - injective module (briefly; a-p-injective module) if every -homomorphim from an annihilator mall principal ub module of to, can be extended to an -homomorphim from to The commutative diagram i 0 m (An -module i a-p-injective module i aid to be a-pq-injective module) F F Example2 Let, where F i a field F F 0 0 F Then i not -injective module i -principal injective module F F Example22 Let, where F i a field F 0 Then i a-mall F F principal--injective module 0 Proof It i clear that X, i the a-mall principal ub module of Let : X be φˆ an -homomorphim ince exit x2, x22 F uch that Then x 2 x 22 0 X, there φ φ, x 2 x22 x 22 it follow that x 2 0 Define φˆ : by 0 0 φˆ, x 22 it i clear that φˆ i an -homomorphim Then φˆ 0 φˆ x Thi how that φˆ i an extenion of φ // x 22 Lemma2 Let be an a-p--injective module with l ( m) l ( n) m, n( 0) Then m n Proof Clear Theorem2 Let m be an -module A module i an a-p-injective module if only if for homomorphim, β : m with i monomorphim Then there exit uch that = Proof Let be an a-p-injective module We define homomorphim, β : m i monomorphim Conider the diagram: If xl ( m) then 0 m (β m)x 0 β(mx) 0 mx kerβ mx 0 ( i monic) o (mx) 0 xl (m) Thu l (β m) l (m) (m) β(m) Then there exit uch that (m) γ(β m) γβ Let β : m be the monic incluion in Then there exit uch that, the following diagram i commute ow 0 m γ(mr) γ(β(mr) γβ(mr) (mr) Thu extend Hence i a-p-injective module// Lemma22 Any direct umm of an a-p-injective module i again a-p--injective module Proof clear Wadbudhe IJEIT Volume 3 Iue 7 July 205 Page 462

3 IJEIT// Vol03 Iue 07//July//Page o:46-465//i x 205 Theorem22 Let m be a-mall principal ub module of with l l (m ) a m Then the following condition are equivalent: e i) l (m) ii) m a Proofi)ii) Let l l (m ) be a-mall principal ubmodule of Therefore l l (m) + X = Then l () l [ l l (m) X] l (m) l (X), o 0 e l (X) 0 ince l (m) [, Lemma 23] m l l (m) Thi implie m a ii)i) l[ l( m)] b ll(m) l(b) hold for every a-mall principal ubmodule m of all b To prove l (m) or an element b Then l (m) (b) l l o l (b) l 0 ince l l (m) a Hence b = 0, becaue l l (b) l (m) a // b i Lemma23 The direct umm of an a-p-injective module i a-p--injective module Proof Clear Lemma2,4 Let ( i n) be a-p-injective module Then n i i alo a-p-injective module Proof It i enough to prove the reult for n = 2 Let m with m i a-mall principal ubmodule of : m 2 be an - homomorphim ince are a-p-injective module there exit -homomorphim : 2 : 2 uch that i π 2i π2, where 2 are projection map from 2 to 2 repectively, i :m i the projection map Put ˆ i i2 2 : 2, where i i2are injection map from to 2 repectively, thu it i clear that ˆ extend // i b Theorem23 Let be -module Then i a-p--injective module if only if for each m with m a Hom (,)m l l (m) Proof Let be a-p--injective module let m be a-mall principal ubmodule of Then l (m) l X 0 l () l [ l l (m X)] l l l (m) l ( X ) o l (X) 0 ince l (m) e By lemma 23 [8] Implie x a Clearly Hom (,)m l l (m) ow let x l l (m) define a map : m x by (mr) xr for every r Then i well defined, In fact mr mr2 for all r,r2 mr mr2 0 m(r r2 ) 0 (r r2 ) l () Hence x(r r2 ) 0 xr xr 2 0 ince i a-p--injective module Therefore extend to ˆ ow x (m) ˆ(m) Hom (m) If r (x) e, m e, where x J () Then m r (x) 0, 0 m it follow that l (0) l (m r (x)) l (m) l ( r (x)) o ( r (x)) 0 Hence m a Let : m be an - homomorphimthen aumption l (m) l r (m) o by (m) ˆ (m) foe ome Hom (,) Thi how i a-p-injective module// Theorem24 The following condition are equivalent for a emi projective module i) Every a-mall principal ubmodule of i projective ii) Every factor module of an a-p-injective module i a-p-injective module iii) Every factor module of an injective module i a-p-injective module Proof [5] Wadbudhe IJEIT Volume 3 Iue 7 July 205 Page 463

4 IJEIT// Vol03 Iue 07//July//Page o:46-465//i x 205 Lemma:24 [8] Conider the following condition for a right -module φ: i) () a ii) t () t(), 0 t iii) l ( ) 0 t iv) l ( ) 0 t v) l(k ) l(k) t Proof: i)ii) Aume that t t() t() Let m, then t (m' ) t(m) for ome m r (t) () ince () ie a r (t) (), r (t) i ubmodule of implie that l (t) r 0 We have t l r (t), thu t 0 ii)iii) Let t l ( ) t( ) 0 t t t(m) t (m) t (m) by (ii) t 0 ie l ( ) 0 iii)iv) Let t l ( ) t( ) 0 t( ) t( ) 0 t 0 iv)v) ie l ( ) 0 t l( ) by (iv) t 0 t l ( ) The other incluion alway hold v)i) Let () X for ubmodule x of Let t (X), m xx, Then t t() l t(m) - t(m') 0 t((m)- (m')) 0 (m) -(m') r (t) mr (t) (m' ) or o t( -(m) ) t(- (m)) t 0 t ker t 0 Hence l (X) 0 Ie () a // Propoition2 Let be a left -module, X y be ub et of with A B be ub et of Then l (X) i a left ideal of X Y l (X) l (Y) A B r (A) r (B) X l r (X) A rl (A) Proof [9] Lemma25 The following tatement are equivalent for an - module : i an a-pq-injective l r (m) med r (m) r (n) where m ed n γ : m, med i an - homomorphim then γ(m) m n m Proof [6] Theorem25 Let be an a-pq-injective module with End (), then l ( r ( ) m) l (m) where m ed(), Proof Let be an a-pq-injective module It i clear that l (m) l ( r ( ) m) Let l ( r ( ) m) If r r( m) ie (m) 0 (m)r 0 mr ( r ( ) m) Then (mr) 0 (m)r 0 rr (β m) Thu r m) r (β m) ince med(), ( m ed(), [] Hence β(m) ( (m)) β(m) ( (m)) 0 (β )(m) 0 β γ l Thu () βγ l () l ( r ( ) m) l (m) // Theorem26 Let be an -module End () Then the following are equivalent: i a-pq-injective l (ker ) with () a ker kert,t with () a t l ( ker Im(t)) l (Imt),t with () a If : (), with () a, then Proof i) ii) [6] Wadbudhe IJEIT Volume 3 Iue 7 July 205 Page 464

5 IJEIT// Vol03 Iue 07//July//Page o:46-465//i x 205 ii)iii) If ker kert, where,t with () a, Then () X, X i ubmodule of, implie l (X) 0 ince () e l kert) l (ker) with (kert ) by (ii) we ( have t l l ( kert) o t iii)iv) Let,t with () a let x (ker Im(t)) x(ker Im(t)) 0 l ker(t) kerxt ince t() e t() a Thuxt t We write xt yt where y Then xt yt 0 (x y)t 0 (x y) l (Imt) Thi how that x l (Imt) The other incluion i clear iv)v) Put t in (iv), we have l (ker) l (ker Im( )) l (Im( )) v)i) Let, with () a Define :() by ((m)) m m // EFEECE ickolon W K, Yiif F, Principally-injective ring, J Algebra 74 (95) ickolon W K, Yiif F, Principally Quai-injective module, Comm Algebra (4) hen L Chen JL, mall injective ring, arviv, A/ Vol-2 (2005) 4 F W Anderon, K Fuller, ing Categorie of odule, pringer- Verlag, ew- York, Yardorn, WQ-Principally injective module, Int J ath Archive, 6(2) 205, Wangwai, mall PQ-peincipallyinjective module Int J ath Archive, 3(3) 202, Wangwai, PQ-injective module, Int J Contemp ath cience, Vol 6 (203), Kalati T A Kekin D, Annihilator-mall ub module, Bull of the Iranian ath oc Vol 39 o 6 (203) F W Anderon, K Fuller, ing Categorie of odule, pringer- Verlag, ew-york, 992 Wadbudhe IJEIT Volume 3 Iue 7 July 205 Page 465