# Canonical Ring Homomorphism

by Dragonfall
Tags: canonical, homomorphism, ring
 P: 994 1. The problem statement, all variables and given/known data f:A->A/I is a ring homomorphism. Does f^-1 take maximal ideas of A/I to maximal ideals of A? 3. The attempt at a solution I think it does, since there is a bijection between A and A/I preserving subsets-ordering. But f might not be that bijection.
 P: 244 There's no bijection between R and R/I unless I is {0} ;) Are you supposed to prove this or do you just want to know? EDIT: Unless of course, R is infinite *blushes* And even then, it won't always be the case.
 P: 994 I mean bijection between ideals of A/I and those of A containing I.
P: 244

## Canonical Ring Homomorphism

Yeah, that's right
 P: 994 So how do I use that to show that f^-1 takes maximal ideas of A/I to those of A?
 P: 244 Well if you're allowed to use the fact that f induces a bijection between the ideals of A containing I and the ideals of A/I that preserves inclusion, then that should be easy. Think about a maximal ideal $$M_{A}$$ containing $$f^{-1}(N_{A/I})$$ where $$N_{A/I}$$ is maximal in A/I.
 P: 994 Do you mean that f *is* a bijection between the ideals of A containing I and the ideas of A/I that preserves inclusion? Not sure what you meant by "induces". Do you mean defining g which acts on the power set of A, and g(x) is the image f(x)?
 P: 244 No because f isn't a map on the ideals! It's a map on elements of A onto cosets of I. I guess we were being a little sloppy earlier. You can think of it like that but formally, they are two distinct maps. It's common to abuse notation and write them the same however.
 P: 994 Ok, I think I got it. Thanks!

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