
#1
Sep2409, 08:02 PM

P: 992

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 subsetsordering. But f might not be that bijection. 



#2
Sep2409, 08:19 PM

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. 



#4
Sep2409, 08:36 PM

P: 244

Canonical Ring Homomorphism
Yeah, that's right




#5
Sep2409, 08:48 PM

P: 992

So how do I use that to show that f^1 takes maximal ideas of A/I to those of A?




#6
Sep2409, 09:02 PM

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 [tex]M_{A}[/tex] containing [tex]f^{1}(N_{A/I})[/tex] where [tex]N_{A/I}[/tex] is maximal in A/I.




#7
Sep2409, 09:07 PM

P: 992

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)?




#8
Sep2409, 09:09 PM

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.



Register to reply 
Related Discussions  
ring homomorphism  Calculus & Beyond Homework  10  
Ring homomorphism  Calculus & Beyond Homework  0  
ring homomorphism  Calculus & Beyond Homework  3  
ring homomorphism  Calculus & Beyond Homework  7  
Ring Homomorphism Question  Linear & Abstract Algebra  3 