prakshepak aavaran

shreni siddhant navachya amoort ganitachya shakhet, 'x' yaa vastuche prakshepak aavaran heey eka arthane prakshepak vastoo 'p' dware 'x' chey sarvottam andaj aahe. prakshepak aavaran heey injection lifafyanche duheri aavaran aahe.
vyakhya.
sootra _ 1 hii eka shreni aanee sootra _ 1 madheel object 'x' asoo dya. eka prakshepak aavaran eka jodee ('p', 'p') aahe, sootra _ 1 madheel 'p' prakshepak object aanee hommadhil ('p', 'x') anavashyak epimorphisms 'p' aahe.
jarr "r" hii eka ring asel, tarr "r"-moduleschya shreneet, anavashyak epimorphism heey nantar epimorphism sootra aahe, jase qee "p" chey curnell heey "p" chey anavashyak up-module aahe.
malmatta.
prakshepak aavarane aanee tyanche anavashyak epimorphism, jevha teey astitvat asatat, tevha teey samarupateparyant adviteey asatat. tathapi, samarupata adviteey asanyachi garaj nahee, kaaran prakshepak gunadharm haah purn swarupacha sarvatrik gunadharm nahee.
anavashyak curnell asalele 'pii' chaa mukhya parinam khalilpramane aaheh jarr 'aine' heey 'pii' chey konatehi yogya upghatak asel, tarr sootra _ 5. anaupacharikpane sangayache tarr, heey darshavite qee anavashyak karnalmule 'pii' laa 'emm' changlya prakare jhakale jaate, mhanje 'pii' chey konatehi upghatak purese honar nahee. heey 'pii' chya prakshepakatela avalambun naheeh heey serve anavashyak epimorphism babatit kharey aahe.
jarr ("pii", "pii") heey "emm" chey prakshepak aavaran asel aanee "pii" heey epimorphism sootra _ 6 asalele aanakhi eka prakshepak module asel, tarr tethe eka vibhajan epimorphism α aahe, jeey "pii" pasoon "pii" paryant aahe, asey sootra _ 7 aahe.
pratyek davya (ujavya) aar-modulesathi astitvat asalelya injective lifafa aanee sapat kavharchya ulat, ring "aar" chee parva naan karata, davya (ujavya) aar-modulemadhye sarvasadharanpane projective cover nasatat. jarr pratyek davya (ujavya) aar-modulemadhye aar-modd (modd-aar) madhye prakshepak aavaran asel tarr ring 'aar' laa left (rait) perfect mhanatat.
jarr pratyek maryaditpane tayar jhalelya davya (ujavya) "aar"-modulemadhye "aar"-modd (modd-aar) madhye prakshepak aavaran asel tarr ringla ardh-paripoorn mhanatat. "ardh-paripoorn" haah davikadil-ujavya samamiteey gunadharm aahe.
jarr idempotents "r"/"j" varoon "r" varr uchalat asateel tarr ringla "lift/radd" mhanatat, jethe "j" haah "r" chaa jacobson radical aahe. lift/reed asanyacha gunadharm prakshepak aavarananchya drushtine darshavila jaau shakatoh jarr aanee fakt jarr 'aar' module 'aar'/'jeey' chya thet bereejmadhye (ujavya kinwa davya modulechya rupat) prakshepak aavaran asel tarach 'aar' mhanje lift/reed.
udaharane.
aar" moduleschya shrenimadhyeh

