
shrenibaddh ring

ganit mein, vishesh roop seey amoort beejganit mein, ack shrenibaddh valay ack aisaa valay haye kii antarnihit yogaatmak samooh abelian samoohon key sootra _ 1 kaa ack pratyaksh yog haye jaise kii sootra _ 2. soochkaank samooh aamtaur parr gair-nakaratmak purnankon yaa purnankon kaa samooh hotaa haye, lekin koee bhee monoid hoe sakata haye. pratyaksh yog apghatan koo aamtaur parr gradation yaa grading key roop mein sandarbhit kiya jaataa haye.
ack shrenibaddh module koo isi tarah paribhaashit kiya gayaa haye (sateek paribhasha key liye neeche dekhein). yeh shrenibaddh sadish rikt sthaan koo saamaanyikrit kartaa haye. ack shrenibaddh module joe ack shrenibaddh ring bhee haye, usey shrenibaddh beejganit kahaa jaataa haye. ack shrenibaddh valay koo shrenibaddh sootra _ 3-beejganit key roop mein bhee dekhaa jaa sakata haye.
ack shrenibaddh valay kii paribhasha mein saha-sakriyata mahatvapurna naheen haye (vaastav mein bilkul bhee upayog naheen kiya jaataa haye); isaliye, yeh dhaarana gair-sahayogi algebra parr bhee lagu hoti haye; ee.jee., koee bhee ack shrenibaddh jhooth beejganit parr vichaar curr sakata haye.
pahali sampatti.
aam taur parr, ack shrenibaddh valay key soochkaank samooh koo gair-nakaratmak purnankon kaa samooh maanaa jaataa haye, jaba taka kii anyatha spasht roop seey nirdisht naa kiya jaae. iss lekh mein yahee mamla haye.
ack shrenibaddh valay ack aisaa valay haye joe ack pratyaksh yog mein vighatit hotaa haye.
seey 
yogaatmak samooh, jaise kii 
sabhi gair-nakaratmak purnank sootra _ 6 aur sootra _ 7 key liye.
sootra _ 8 kaa ack gair-shoonya tatva koo "degree" sootra _ 7 kaa "sajaatiya" kahaa jaataa haye. ack pratyaksh yog kii paribhasha key anusaar, sootra _ 11 kaa pratyek gair-shoonya tatva sootra _ 10 koo ack yog sootra _ 12 key roop mein advitiya roop seey likhaa jaa sakata haye jahaan pratyek sootra _ 13 yaa tou 0 haye yaa degree sootra _ 14 kaa sajaatiya haye. gair-shoonya sootra _ 13 sootra _ 10 key "sajaatiya ghatak" hain.
kuchh buniyaadi guna hainh
ack aadarsh sootra _ 26 "sajaatiya" haye, yadi pratyek sootra _ 27 key liye, sootra _ 10 key sajaatiya ghatak bhee sootra _ 29 seey sambandhit hain (samkaksh roop seey, yadi yeh sootra _ 11 kaa ack shrenibaddh upa-module haye; dekhein.) sootra _ 8 key saath ack sajaatiya aadarsh sootra _ 31 kaa pratichhedan sootra _ 8 kaa ack sootra _ 17-upa-module haye jise sootra _ 31 key degree sootra _ 7 kaa "sajaatiya bhaag" kahaa jaataa haye. ack sajaatiya aadarsh isake sajaatiya bhaagon kaa pratyaksh yog haye.
yadi sootra _ 31 sootra _ 11 mein ack dou-tarfa sajaatiya aadarsh haye, tou sootra _ 39 bhee ack shrenibaddh valay haye, jise vighatit kiya jaataa haye.
jahan sootra _ 41 sootra _ 31 key degree sootra _ 7 kaa sajaatiya bhaag haye.
shrenibaddh module.
module siddhant mein sambandhit vichaar ack shrenibaddh module kaa haye, arthaat ack shrenibaddh ring "aar" parr ack baaen module "aim" jaise kii yeh bhee kii
aur
udaaharan key liyeh ack shrenibaddh sadish sthaan ack kshetra parr shrenibaddh module kaa ack udaaharan haye (kshetra mein tuchchh shrenikaran key saath).
udaaharan key liyeh ack shrenibaddh valay apane aap mein ack shrenibaddh module haye. ack shrenibaddh valay mein ack aadarsh sajaatiya haye yadi aur kewal tabhi jaba yeh ack shrenibaddh upa-module hoe. ack shrenibaddh module kaa vinashkarta ack sajaatiya aadarsh haye.
udaaharan key liyeh ack commutative ring "r" mein ack aadarsh "i" aur ack "r"-module "m" diye jaane parr, pratyaksh yog sootra _ 53 sambaddh graded ring sootra _ 54 parr ack graded module haye.
graded module key beach ack morphism sootra _ 55, jise graded morphism kahaa jaataa haye, antarnihit module kaa ack morphism haye joe grading kaa sammaan kartaa haye; i.ee.sootra _ 56. ack shrenibaddh upa-module ack upa-module haye joe apane aap mein ack shrenibaddh module haye aur aisaa kii sett-saiddhaantik samavesh shrenibaddh module kaa ack rupankan haye. spasht roop seey, ack shrenibaddh module "aine" "aim" kaa ack shrenibaddh upa-module haye yadi aur kewal tabhi jaba yeh "aim" kaa upa-module haye aur sootra _ 57 koo santusht kartaa haye. kernell aur shrenibaddh module key rupankan kii chhavi shrenibaddh upa-module hain.
tippanih ack shrenibaddh aakriti koo ack shrenibaddh valay seey dusare shrenibaddh valay mein denaa, jisamen kendra mein padee chhavi key saath, baad vaale valay koo shrenibaddh beejganit kii sanrachana dene key samaan haye.
ack shrenibaddh module sootra _ 58 diye jaane parr, sootra _ 58 kaa sootra _ 59-twist sootra _ 61 dwara paribhaashit ack shrenibaddh module haye. (cf. beejganitiya jyaamiti mein sair kaa mudataa sheef.)
m" aur "n" koo shrenibaddh module honey dein. yadi sootra _ 62 module kaa ack rupankan haye, tou "f" koo degree "d" kahaa jaataa haye yadi sootra _ 63. vibhedak jyaamiti mein vibhedak roopon kaa ack baahari vyutpann degree 1 vaale iss tarah key rupankan kaa ack udaaharan haye.
shrenibaddh module key aparivartaniya.
ack shrenibaddh module "m" koo ack commutative shrenibaddh ring "r" parr diyaa gayaa haye, koee bhee aupchaarik shakti shrinkhala sootra _ 64 koo jodd sakata haih
(sootra 66 koo seemit maanate hue.) ise "aim" kii hilbert-pointcare shrinkhala kahaa jaataa haye.
yadi antarnihit module koo parimit roop seey utpann kiya jaataa haye tou ack shrenibaddh module koo parimit roop seey utpann kiya jaataa haye. generator koo sajaatiya maanaa jaa sakata haye (generator koo unake sajaatiya bhaagon seey badalkar.)
maan leejiye kii "r" ack bahupad ring sootra haye _ 67, "k" ack kshetra, aur "m" isake oopar ack parimit roop seey utpann shrenibaddh module haye. phir falan sootra _ 68 koo "m" kaa hilbert falan kahaa jaataa haye. falan badey "aine" key liye purnank-moolya vaale bahupad key saath mell khaata haye jise "aim" kaa hilbert bahupad kahaa jaataa haye.
shrenibaddh beejganit.
ack beejganit "a" ack valay parr "r" ack shrenibaddh beejganit haye yadi ise ack valay key roop mein shrenibaddh kiya jaataa haye.
samanya mamle mein jahaan ring "r" koo shrenibaddh naheen kiya jaataa haye (vishesh roop seey yadi "r" ack kshetra haye), tou ise tuchchh shrenikaran diyaa jaataa haye ("r" kaa pratyek tatva degree 0 kaa hotaa haye). iss prakaar, sootra _ 69 aur shrenibaddh tukade sootra _ 70 "r"-module hain.
usa mamle mein jahaan ring "aar" bhee ack shrenibaddh ring haye, tou ack kii aavashyakta haye kii 
dusare shabdon mein, hamein "a" koo "r" parr ack shrenibaddh baaen module honey kii aavashyakta haye.
ganit mein shrenibaddh algebra key udaaharan aam hainh
shrenibaddh algebra kaa upayog commutative beejganit aur beejganitiya jyaamiti, samroop beejganit aur beejganitiya topology mein kiya jaataa haye. ack udaaharan sajaatiya bahupadon aur prakshepya kismon (see. aff.) key beach ghanishth sambandh haye. sajaatiya nirdeshank valay.)
"jee"-shrenibaddh valay aur algebra.
uparokt paribhashaon koo soochkaank samooh key roop mein kissy bhee monoid "jee" kaa upayog karke shrenibaddh kiye gaye rings key liye saamaanyikrit kiya gayaa haye. ack "jee"-shrenibaddh valay "aar" ack seedhi raashi apghatan key saath ack valay haye.
kii
r" key tatva joe kuchh sootra _ 81 key liye sootra _ 1 key ander hotey hain, unhein grade "i" kaa sajaatiya kahaa jaataa haye.
shrenibaddh valay" kii pahale seey paribhaashit dhaarana abb ack sootra _ 82-shrenibaddh valay key roop mein samaan hoe jaati haye, jahan sootra _ 82 yog key tahat praakritik sankhyaon kaa ekmusht haye. shrenibaddh module aur algebra kii paribhashaon koo iss tarah seey kissy bhee monoid "jee" key saath anukramanika sett sootra _ 82 koo pratisthaapit kiya jaa sakata haye.
tippanih
udaaharanonh
pratisamvedansheelta.
kuchh shrenibaddh valay (yaa algebra) ack pratisanyojak sanrachana key saath sampann hotey hain. iss dhaarana key liye dou tatvon key saath sootra _ 86 key yogaatmak monoid mein gradation key monoid key samroopata kii aavashyakta hoti haye. vishesh roop seey, ack hastaaksharit monoid mein ack jodee sootra _ 87 hotaa haye jahan sootra _ 88 ack monoid haye aur sootra _ 89 yogaatmak monoids kaa ack samroopata haye. ack pratisanyojak sootra _ 88-shrenibaddh valay ack valay "a" haye jise γ key sambandh mein shrenibaddh kiya jaataa haye jaise kih
sabhi sajaatiya tatvon "x" aur "y" key liye.
shrenibaddh monoid.
sahaj roop seey, ack shrenibaddh monoid ack shrenibaddh ring kaa subset haye, sootra _ 97, sootra _ 8 dwara utpann, yogaatmak bhaag kaa upayog kiye binaa. yaani, shrenibaddh monoid key tatvon kaa samooh sootra _ 99 haye.
aupchaarik roop seey, ack shrenibaddh monoid ack monoid sootra _ 100 haye, jisamen ack shrenikaran falan sootra _ 101 haye jaise kii sootra _ 102. dhyaan dein kii sootra _ 103 kaa shrenikaran aavashyak roop seey 0 haye. kuchh lekhakon ney aage usa sootra _ 104 kaa anurodh kiya haye.
jaba "m" pehchaan naheen haye.
gair-pehchaan tatvon key gradation koo gair-shoonya maanate hue, gradation "aine" key tatvon kii sankhya adhikatam sootra _ 105 parr haye jahan "jee" monoid key utpann karane vaale sett "jee" kii cardinality haye. isaliye gradation "aine" yaa usase cumm key tatvon kii sankhya adhikansh sootra _ 106 (sootra _ 107 key liye) yaa sootra _ 108 haye. vaastav mein, aisaa pratyek tatva "jee" key adhikansh "aine" tatvon kaa utpaad haye, aur kewal sootra _ 108 aise utpaad maujood hain. isi tarah, pehchaan tatva koo dou gair-pehchaan tatvon key utpaad key roop mein naheen likhaa jaa sakata haye. yaani, iss tarah key shrenibaddh monoid mein koee ikai vibhaajak naheen haye.
ack shrenibaddh monoid dwara anukramit shakti shrinkhala.
yeh dhaaranaaen shakti shrinkhala valay kii dhaarana koo vistaarit karane kii anumati detee hain. anukraman parivaar key sootra _ 110 honey key bajaay, anukraman parivaar koee bhee shrenibaddh monoid hoe sakata haye, yeh maanate hue kii pratyek purnank "aine" key liye degree "aine" key tatvon kii sankhya seemit haye.
adhik aupchaarik roop seey, sootra 111 koo ack manmana ardh-nirdhaaran aur sootra 112 koo ack shrenibaddh monoid honey dein. phir sootra _ 113 "k" mein gunank key saath shakti shrinkhala key ardhanirdhaaran koo "r" dwara anukramit kartaa haye. isake tatva "r" seey "k" taka key kaary hain. dou tatvon key yog sootra 114 koo bindu kii disha mein paribhaashit kiya gayaa haye, yeh sootra 115 koo sootra 116 parr bhejane vaalaa kaary haye, aur utpaad sootra 115 koo anant yog sootra 118 parr bhejane vaalaa kaary haye. yeh yog sahee dhang seey paribhaashit kiya gayaa haye (i.ee., parimit) kyonki, pratyek "m" key liye, kewal ack seemit sankhya mein jode hain.
udaaharan key liye.
aupchaarik bhasha siddhant mein, ack varnamala "aeye" diye jaane parr, "aeye" parr shabdon key mukt monoid koo ack shrenibaddh monoid key roop mein maanaa jaa sakata haye, jahaan ack shabd kaa gradation isaki lambai haye.

