286 6. RINGS '.x/ C I is a ring isomorphism of R=I onto R= I . Equivalently, .R=J/=.I=J/ Š R=I as rings. Proof. By Proposition 2.7.13 , the map x C I 7! '.x/ C I is a group isomorphism from .R=I; C / to . R= I; C / . But the map also respects mul-tiplication, as .x C I/.y C I/ D xy C I 7! '.xy/ C I D .'.x/ C I/.'.y/ C I/: We can identify R with R=J by the homomorphism theorem for rings, and this identiﬁcation carries I to the image of I in R=J , namely I=J . Therefore, .R=J/=.I=J/ Š R= I Š R=I: n Proposition 6.3.9. (Factorization Theorem for Rings) Let ' W R ! R be a surjective homomorphism of rings with kernel I . Let J ± I be an ideal of R , and let ± W R ! R=J denote the quotient map. Then there is a
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