mukt bijganit

ganitamadhye, visheshtah ring theary mhanoon olakhalya olakhalya amoort bijganitachya kshetrat, mukt bijganit heey bahupadi ringche gair-sampreshaniy analogue aahe kaaran tyachya ghatakanche varnan gair-sampreshaniy chalansah "bahupadi" mhanoon kele jaau shakate. tyachpramane, bahupadi ringla mukt parivartaneey bijganit manale jaau shakate.
vyakhya.
r" yaa parivartaneey ringsathi, "n" vareel mukt (sahyogi, eksandh) bijganit anishchit {"x" 1..xn", haah mukt "r"-module aahe, jyacha aadhar {"x" 1 varnamalavaril serve shabdancha samavesh aahe.., "xn"} (rikt shabdasah, joou mukt bijganitacha ekak aahe). gunakar khalilpramane paribhashit karoon heey "r"-module "r"-bijganit banateh done aadhar ghatakanche gunakar mhanje sambandhit shabdanche sanyojanh
aanee asha prakare done aniyantrit "aar"-module ghatakanche gunakar advitiyapane nirdharit kele jaate (kaaran "aar"-bijganitetil gunakar "aar"-dvimiteey asane aavashyak aahe). heey "r"-bijganit "r" λ "x" 1 darshavile jaate.., "xn" γ. hii rachana sahajpane anishchitanchya aniyantrit sanch 'x' madhye samanyikrut kelee jaau shakate.
thodakyat, aniyantrit sanch sutrasathi _ 2, mukt (sahyogi, eksandh) "r"-bijganit "x" varr aahe 
r"-dvimiteey gunakarasah jeey shabdanvar sanyojan aahe, jethe "x" * heey "x" (i) varr mukt monoid darshavate.eae. x" i) yaa aksharanvaril shabd, sootra _ 4 hii bahya thet berries darshavate aanee "rw" heey 1 ghatkavaril mukt "r"-module, mhanje "w" haah shabd darshavate.
udaharanarth, 'r' λ 'x' 1, 'x' 2, 'x' 3, 'x' 4 'madhye, scalar' α, β, γ, δ 'ω' r 'sathee, done ghatakanchya gunakarache thos udaharan aaheh 
sootra _ 5.
gair-vinimaysheel bahupadi ring hii 'x' i 'madheel serve maryadit shabdanchya mukt monoidchya' r 'vareel monoid ringsah olakhali jaau shakate.
bahupadishi virodhabhas.
kaaran varnamalavaril shabd {"x" 1,..., "xn"} haah "r" λ "x" 1 chaa aadhar banato.., xn" ng, heey spasht aahe qee "r" λ "x" 1 chaa konatahi ghatak,...xn" γ heey yaa swarupat advitiyapane lihile jaau shakateh
jethe sootra _ 7 heey "r" chey ghatak aahet aanee yapaiki serve parantu maryaditpane anek ghatak shoonya aahet. r" λ "x" 1 chey ghatak qaa aahet heey yavarun spasht hote.., xn" ω laa anekada "chull" (kinwa "anishchit") "x" 1 madhye "gair-vinimaysheel bahupadi" mhanoon darshavile jaate.., xn"; ghatak sootra _ 8 yaa bahupadiche "gunank" aanee "r"-bijganit "r" λ "x" 1 asey mhatale jaate.., xn" γ laa "" n "" anishchitanmadhye "" r "varr" "gair-samparivartak bahupadi bijganit" "mhanatat". lakshat ghya qee vastavik bahupadi ringchya ulat, chull badalat naheet. udaharanarth, "x" 1 "x" 2 mhanje "x" 2 "x" 1 barobar nahee.
adhik samanyatah, janitranchya konatyahi sanch 'eae' varr mukt bijganit 'aar' λ 'eae' λ tayar karata yete. ringjala zed-algebras manale jaau shakate, tyamule "eae" vareel mukt ringchi vyakhya mukt bijganit zed τ "eae" λ mhanoon kelee jaau shakate.
kshetravar, 'aine' anishchitanvar mukt bijganit 'aine'-aayami vector jaaget tensor bijganit mhanoon tayar kele jaau shakate. adhik samanya gunank ringsathi, jarr aapan "aine" generatorvar mukt module ghetale tarr teech rachana karya karate.
eae" varr mukt bijganitachi rachana hii karyatmak swarupachi aahe aanee yogya sarvatrik malmatteche samadhan karate. mukt bijganit karyavahak haah 'aar'-algebraschya shrenipasoon sanchanchya shreniparyantchya visarnara karyavahak yachya shejari thewala jaato.
vibhajak ringjavareel mukt algebra heey mukt aadarsh rings aahet.

