DOI: /jam.v14i2.7401

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1 Nutrosoph Soft oduls Kml Vlyv Sd Byrmov Dprtmnt of Algbr nd Gomtry of Bku Stt Unvrsty ZKhllov str AZ48 Bku Azrbjn Abstrt olodtsov nttd th onpt of soft sts n [7] j t l dfnd som oprtons on soft sts n [] Aktş t l gnrlzd soft sts by dfnng th onpt of soft groups n [] Aftr thn Qu- Sun t l gv soft moduls n [0] n ths ppr th onpt of nutrosoph soft modul s ntrodud nd som of ts bs proprts r studd Ky Words Nutrosoph st nutrosoph soft st nutrosoph soft moduls nutrosoph soft homomorphsm Dt of Publton 0508 DO 0497/jmv4740 SSN 47-9 Volum 4 ssu 0 Journl Journl of Advns n thmts Wbst https//rworldom ntroduton hs work s lnsd undr Crtv Commons Attrbuton 40 ntrntonl Lns h ontrbuton of mthmts to th prsnt-dy thnology n rhng to fst trnd nnot b gnord h trors prsntd dffrntly from lssl mthods n studs suh s fuzzy st []ntutonst fuzzy st [] soft st [7] nutrosoph st [9] t h lgbr strutur of st thors dlng wth unrtnts hs lso bn studd by som uthorsaftr olodtsov s work som dffrnt ppltons of soft sts wr studd n [6] j t l [4] prsntd th onpt of fuzzy soft st Rosnfld [8] proposd th onpt of fuzzy groups n ordr to stblsh th lgbr struturs of fuzzy sts Aktş nd Çğmn [] dfnd soft groups nd omprd soft sts wth fuzzy sts nd rough sts Aftr th dfnton of fuzzy soft group s gvn by som uthors [4] ng t l [8] gv soft smrngs nd UAr t l [] ntrodud ntl onpts of soft rngs Dfnton of fuzzy modul s gvn by som uthors [] Qu- Sun t l [0] dfnd soft moduls nd nvstgtd thr bs proprts uzzy soft moduls nd ntutonst fuzzy soft moduls ws gvn nd rsrhd by C Gunduz (Ars nd S Byrmov [90] h mn purpos of ths ppr s to ntrodu bs vrson of nutrosoph soft modul thory whh tnds th noton of modul by nludng som lgbr struturs n soft sts nlly w nvstgt som of nutrosoph soft modul bs proprts 7670

2 Prlmnrs n ths ston w wll gv som prlmnry nformton for th prsnt study Dfnton [9] A nutrosoph st A on th unvrs of dsours s dfnd s A _ + + whr 0 nd Dfnton [7] Lt b n ntl unvrs b st of ll prmtrs nd P ( dnots th powr st of A pr ( s lld soft st ovr whr s mppng gvn by P( rstly nutrosoph soft st dfnd by j [] nd ltr onpt hs bn modfd by Dl nd Brom [7] s gvn blow Dfnton Lt b n ntl unvrs st nd b s st of prmtrs Lt P ( dnot th st of ll nutrosoph sts of hn nutrosoph soft st ( dfnd by st vlud funton rprsntng mppng P( ppromt funton of th nutrosoph soft st ( ovr s st whr s lld n othr words th nutrosoph soft st s prmtrzd fmly of som lmnts of th st P ( nd thrfor t n b wrttn s st of ordrd prs ( ( rsptvly lld th truth-mmbrshp ndtrmny- whr 0 mmbrshp flsty-mmbrshp funton of nqulty Dfnton 4 [6] Lt ( omplmnt of ( Obvous tht ( Sn suprmum of h s so th s obvous b nutrosoph soft st ovr th ommon unvrs ( s dnotd by ( nd s dfnd by ( ( ( Dfnton 5 [] Lt ( nd ( G unvrs ( ( s sd to b nutrosoph soft subst of ( G b two nutrosoph soft sts ovr th ommon G G G G f h t s dnotd by 767

3 767 h oprtons of unon ntrston dffrn AND OR on nutrosoph soft sts r dfnd dffrntly from th studs [6] n ddton bs proprts of ths oprtons wll b prsntd Dfnton 6 Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn thr unon s dnotd by nd s dfnd by whr mn m m Dfnton 7 Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn thr unon s dnotd by nd s dfnd by whr m mn mn Dfnton 8 Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn dffrn oprton on thm s dnotd by \ nd s dfnd by s follows whr m mn mn Dfnton 9 Lt b fmly of nutrosoph soft ovr th ommon unvrs hn

4 767 nf sup sup sup nf nf Dfnton 0 Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn AND oprton on thm s dnotd by nd s dfnd by whr m mn mn Dfnton Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn OR oprton on thm s dnotd by nd s dfnd by whr mn m m Dfnton ( A nutrosoph soft st ovr s sd to b null nutrosoph soft st f ; 0 0 t s dnotd by 0 ( A nutrosoph soft st ovr s sd to b bsolut nutrosoph soft st f 0; t s dnotd by Clrly 0 nd 0 Proposton Lt nd b two nutrosoph soft sts ovr th ommon unvrs hn

5 ( ( ( ( ( ( ( nd ( ( ( ( ( ( ; ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ; ( ( 0( ( nd ( 0( 0( ; (4 ( ( ( nd ( ( ( nd Proposton Lt ( nd ( unvrs ( hn ( ( ( ( ; Proposton Lt ( nd ( ( hn ( ( ( ( ; b two nutrosoph soft sts ovr th ommon Dfnton Lt b lft R -modul nd lt A ( hn w sy ( b two nutrosoph soft sts ovr th ommon unvrs b nutrosoph st ovr s nutrosoph modul f th followng ondtons r stsfd ( 0 ( 0 ; ( 0 0 b ( + y ( y ; ( + y ( y ; ( + y m ( y ( ; ; Dfnton 4 Lt ( nd ( nd b two nutrosoph moduls ovr rsptvly W sy tht f s homomorphsm of nutrosoph moduls f th followng f moduls r stsfd ondtons for homomorphsm of ( f ( f ( f Nutrosoph soft moduls n ths ppr R s n ordnry rng Lt b lft (or rght R -modul nd lt NS dnots th fmly of nutrosoph sts ovr Dfnton Lt ( A nutrosoph soft modul ovr ff of nd dnotd s b nutrosoph soft st ovr hn ( A A ( A b st s sd to b s nutrosoph submodul 7674

6 Dfnton Lt ( A nd B rsptvly nd lt b two nutrosoph soft moduls ovr nd N f N b homomorphsm of moduls nd lt g A B b f g A B s nutrosoph soft mppng of sts hn w sy tht homomorphsm of nutrosoph soft moduls f th followng ondton s stsfd ( g Not tht for g f g g f g g f nutrosoph moduls ( A f N g s nutrosoph homomorphsm of Nutrosoph soft moduls nd morphsms of thr s onssts of tgory hs tgory s dnotd NS horm Lt ( A nd B ntrston ( A ( B b two nutrosoph soft moduls ovr hn thr s nutrosoph soft modul ovr Proof Lt ( A ( B ( C whr C A B Sn th nutrosoph soft st s nutrosoph submodul for C s nutrosoph soft modul ovr horm Lt ( A nd B ( A ( B s nutrosoph soft modul ovr Proof W n wrt ( A ( B ( A B submoduls of ( C b two nutrosoph soft moduls ovr hn Sn nd b r nutrosoph b s nutrosoph submodul of hus ( b ( b ( b b b b s nutrosoph submodul of for ll ( b A B Hn w fnd tht ( A ( B horm Lt ( A nd B thn ( A ( B s nutrosoph soft modul ovr b two nutrosoph soft moduls ovr f A B s nutrosoph soft modul ovr Proof W n wrt ( A ( B ( C A B or B A for ll C f A B B A of nd f thn A B s nutrosoph soft modul ovr Sn A B t follows tht thr thn b b s nutrosoph submodul b b s nutrosoph submodul of Hn Dfnton Lt ( A nd B b two nutrosoph soft moduls ovr hn A s lld nutrosoph soft submodul of ( B f A B or ll A ( s nutrosoph submodul of ( 7675

7 b two nutrosoph soft moduls ovr f horm 4 Lt ( A nd A ll A thn ( A s nutrosoph soft submodul of A Proof h proof of th thorm s strghtforwrd horm 5 Lt ( A b nutrosoph soft moduls ovr nd lt ( A nonmpty fmly of nutrosoph soft submoduls of ( A hn ( s nutrosoph soft submodul of ( A A ( A s nutrosoph soft submodul of ( A f A A for ll j j Lt ( A nd B ( f g ( A ( B for thn ( A s nutrosoph soft submodul of ( A b two nutrosoph soft moduls ovr nd N rsptvly nd b nutrosoph soft homomorphsm of ths moduls Now n ths ston w ntrodu th krnl nd mg of nutrosoph soft homomorphsm of nutrosoph soft moduls Lt kr f Dfn A NS( by hn ( A s nutrosoph soft modul ovr t s A lr tht ths modul s nutrosoph soft submodul of Dfnton 4 ( A s sd to b krnl of ( f g nd dnotd by ( f g kr Now lt B g( A hn for ll b B thr sts A suh tht b N m f N W dfn th mppng B NS( N ( b ( g N ( b ( g N ( b ( g N Sn ( g f f f b g Lt s f s s stsfd for ll s nutrosoph soft homomorphsm ( g ( g ( g A hn th pr ( B s nutrosoph soft modul ovr N nd B nutrosoph soft submodul of ( B Dfnton 5 ( B Proposton Lt ( A s sd to b mg of ( f g nd dnotd by ( f g m b nutrosoph soft modul ovr N b n R modul nd f N b homomorphsm of f A s nutrosoph soft modul ovr N Proof f th mppng f A NS( N R moduls hn ( s dfnd by 7676

8 th proof s ompltd Not tht ( f A ( A ( f A moduls ( f ( ( y sup f ( f ( ( y sup f ( f ( ( y nf f y y y s nutrosoph soft homomorphsm of nutrosoph soft Proposton f s n R modul ( A s nutrosoph soft modul ovr N nd f N s homomorphsm of f A s nutrosoph soft modul ovr Proof f th mppng f A NS( ( f ( ( f f ( R moduls thn ( f s dfnd by f f th proof s ompltd t s lr tht ( f A ( f A ( A soft moduls s nutrosoph soft homomorphsm of nutrosoph Lmm Lt nd N b n R moduls nd f N b n R homomorphsm nd A A r two nutrosoph soft moduls ovr nd N rsptvly ( nd ( ( f A ( A ( A f ( f ( f ( s nutrosoph soft homomorphsm f nd only f for ll A s stsfd ( ( f A ( A ( A f ( f ( f ( s nutrosoph soft homomorphsm f nd only f for ll A s stsfd horm 6 f ( A ( A Proof Dfn whr p p s nutrosoph soft modul ovr A s fmly of nutrosoph soft moduls ovr ( by ( p ( ( ( p p s projton mppng Sn ( 0 p ( 0 p ( 0 thn 7677

9 s nutrosoph soft modul ovr lso nutrosoph soft modul ovr horm 7 f ( A thn ( A for ll p ( p p s s fmly of nutrosoph soft moduls ovr th fmly of moduls s nutrosoph soft modul ovr Proof Dfn A ( j ( ( ( j j ( for ll by A whr j s mbddng mppng Sn ( for ll j j j s nutrosoph soft submodul ovr s nutrosoph submodul ovr Lmm Gvn moduls A f N f nutrosoph soft modul A nd N nd fmly of R r nutrosoph soft moduls ovr A ovr N suh tht for ll ( A A f s nutrosoph soft homomorphsm of nutrosoph soft moduls homorphsms thn thr st Gvn moduls nd N nd fmly of R homorphsms B g N ( B r nutrosoph soft moduls ovr N modul A ovr suh tht for ll f thn thr st nutrosoph soft ( B g A s nutrosoph soft homomorphsm of nutrosoph soft moduls Proof Dfn A N ( ( f ( by f ( ( f Dfn A ( by ( g ( ( ( g g for ll A 7678

10 By usng ths lmm w dfn th onpts of submodul quotnt modul produt nd oprodut oprtons n th tgory of nutrosoph soft moduls Corllry f ( A N s mbddng mppng thn ( ( A Corllry f ( A projton thn ( p ( A f ( A s nutrosoph soft modul ovr nd N s submodul of nd s nutrosoph soft modul ovr N s nutrosoph soft modul ovr nd p s nonl s nutrosoph soft modul ovr quotnt modul s fmly of nutrosoph soft moduls ovr th fmly of moduls n dfn th produt nd oprodut of ths fmls by A nd ( A thn w rsptvly horm 8 h tgory of nutrosoph soft moduls hs zro objts sums produt krnl nd okrnl b Lt nd N b rsptvly rght nd lft moduls ovr R (rng Lt ( A nd B two nutrosoph soft moduls ovr nd N rsptvly W onsdr tnsor produt of moduls s N h mppng s dfnd by for ( b A B A B N ( ( b ( b ( ( b ( b ( ( b ( b Dfnton 6 ( A B s sd to b tnsor produt of ( A nd B by ( A ( B horm 9 ( A B Proof or ( b A B s nutrosoph soft modul ovr ( nd ( b N nd ( A B nutrosoph submodul ovr N N N r nutrosoph soft moduls nd dnotd b s s nutrosoph soft modul ovr nd Dfnton 7 ( A B s sd to b tnsor produt of ( A nd B dnotd by ( A ( B 7679

11 Conluson hs ppr summrzd th bs onpts of nutrosoph soft sts nd nutrosoph soft moduls By usng ths onpts w studd th lgbr proprts of nutrosoph soft sts n modul strutur Rfrn U Ar Koyunu Bny Soft sts nd soft rngs Comput th Appl59 ( HAktş N Çğmn Soft sts nd soft group nformton Sn 77 ( Atnssov K ntutonst fuzzy sts uzzy Sts nd Systms 0 ( AAygünoğlu HAygün ntroduton to fuzzy soft groups Comput th Appl58 ( Br hptr N K On nutrosoph soft funton Ann uzzy th nform ( ( Br hptr N K ntroduton to nutrosoph soft topologl sp Opsrh 54(4 ( Dl Broum S Nutrosoph soft rltons nd som proprts Ann uzzy th nform 9( ( ng YB Jun Zho Soft smrngs Comput th Appl56 ( CGunduz (Ars nd S Byrmov uzzy soft moduls ntrntonl thmtl orum Vol 6 0 no CGunduz (Ars nd S Byrmov ntutonst fuzzy soft moduls Computr nd thmts wth Applton 6( LJn-lng YRu- YBng-u uzzy soft sts nf fuzzy soft groups Chns Control nd Dson Confrn ( SRLopz-Prmouth DSlk On tgors of fuzzy moduls nformton Sns 5 (990-0 j P K Nutrosoph soft st Ann uzzy th nform 5( ( PKj RBsms ARRoy uzzy soft sts h Journl of uzzy thmts 9 ( ( PKj RBsms ARRoy Soft st thory Comput th Appl 45 ( PKj ARRoy An Applton of soft sts n dson mkng problm Comput th Appl44 ( D olodtsov Soft st thory-frst rsults Comput th Appl7 ( ARosnfld uzzy groups Journl of thmtl Anlyss nd Appltons 5 (

12 9 Smrndh Nutrosoph st gnrlston of th ntutonst fuzzy sts nt J Pur Appl th 4 ( Qu- Sun Z-Long Zhng Jng Lu Soft sts nd soft moduls Ltur Nots n Comput S 5009 ( Zdh L A uzzy Sts nform Control 8 ( Zhd RAmr On uzzy Projtv nd njtv oduls h journl of uzzy thmts Vol No (

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