Hello I just had a quick question about ring homomorphisms.

Is it possible to have a ring isomorphism from to F, where F is a field but is not a field?

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- Apr 17th 2011, 09:28 AMslevvioQuestion about Ring Homomorphism
Hello I just had a quick question about ring homomorphisms.

Is it possible to have a ring isomorphism from to F, where F is a field but is not a field? - Apr 17th 2011, 09:35 AMabhishekkgp
edit: deleted post.

hi slevvio!!

f: R----> F

f(x)=0 for all x in R.

this is trivial homomorphism i guess. Is this acceptable to you?? - Apr 17th 2011, 09:50 AMslevvio
Thanks but that is not a ring isomorphism!

- Apr 17th 2011, 09:57 AMDeveno
if φ is onto, R has to be a field, since then R = φ^-1(F).

if φ is not onto, and φ(R) is just a subring of F, then sure, consider the identity homomorphism which includes Z in Q. - Apr 17th 2011, 10:08 AMabhishekkgp
- Apr 17th 2011, 10:14 AMDeveno
- Apr 17th 2011, 08:22 PMabhishekkgp
- Apr 19th 2011, 04:19 AMslevvio
Thanks guys. I was just confused because in my notes a ring homomorphism was defined from a ring to a field, and then after this was proved to be a ring isomorphism, he showed that the initial ring was actually a field. But this should arise from the fact we have a ring isomorphism. I would type out the specific example but Latex is borked