Can the Kernel of a Ring Homomorphism Equal 12Z or 13Z?

[AkilMAI]

Let f : Z ->C be a homomorphism of rings. Can the kernel of f be equal to 12Z or 13Z?
Ok,the way I'm thinking about it is using a proof by contradiction:asuming ker f=12Z...then by the First Isomorphism Theorem for rings Z/ker f ~im f where I am f is by definition a subring of C.But since I am f=12Z is not an integral domain and every subring in C is an integral domain the I am f will not be a subring oc C which is a contradiction.
The same thing with 13Z,is not equal with the kernel.

 

[Deveno]

Re: Kernerl and homomorphism

think about what happens to f(1).

if f(12) = 0, then f(12) = f(1 + 1 +...+ 1) = f(1) + f(1) +...+ f(1) = 12f(1) = 0.

since C is an integral domain, and 12 ≠ 0, f(1) = 0. but f(1) = 1, since f is a ring homomorphism.

 

[AkilMAI]

Re: Kernerl and homomorphism

ok either ker f=12Z or ker f= {0}
f(n)=0>n*f(1)=0 but f(1)=1since f is a ring homomorphism.So ker f={0} or

Z/ker f is an integral domain since it is a subring of C =>ker f/=12Z. or ker f=/13
I'm I doing this wrong?

 

[Amer]

Re: Kernerl and homomorphism

f is ring homomorphism which means f(1) =1 must be
but if 12Z is the kernel this will drive us to f(1) =0 which contradict with the ring homomorphism
so 12Z,13Z can't be the kernel, ker(f) = {0}

 

[AkilMAI]

Re: Kernerl and homomorphism

Thank you for the confirmation