Hi,

I was wondering if someone would be able to give me a hint on this problem I am working on. The problem sates: "Iff: A-->B is a homomorphism from A onto B, and B is a field, then the kernel offis a maximal ideal. "

I know that the kernel offare all the elements of A that are carried to the "0" element of B. I also know that an ideal means that it must be "closed with respect to addition, negatives and B absorbs products of A".

So, I think that if I take an elementxin A, and I applyfto it that means that the elementxI started off with will be mapped to "0" which is in B. So the kernf= {f:f(x) = 0}.

Now I can show the following things are true:

1. Closes with respect to addition.f(x) = 0 , and_{1}f(x)=0, then_{2}f(x) =_{1}+x_{2}f(x)+_{1}f(x)=0._{2}

2. Closed with respect to negatives.f(x) = 0,f(-x) = 0 = -f(x) = 0 = -0 = 0.

3. B absorbs products of A. Iff(x) = 0, thenf(x*y) =f(x) *f(y) = 0 *f(y) = 0.

I believe what I showed here is that the kernel offis an ideal, but I don't know how to show it is a maximal ideal.

There was a hint given in the problem. It states that it may help if I : "Show that a field can have no non trivial ideals". I have proven that. But I don't know how that is going to help me with my proof. I also realize I didn't take into consideration that B is a field. Am I missing a step because of that as well?

Any help is greatly appreciated.

Thank you for your time!