Is the map $f: \mathbb C \to \mathbb Z$, $f(a+bi)=a$, for $a,b$ in $\mathbb Z$, a homomorphism of rings?
Any one could give me some ideas please?
Is the map $f: \mathbb C \to \mathbb Z$, $f(a+bi)=a$, for $a,b$ in $\mathbb Z$, a homomorphism of rings?
Any one could give me some ideas please?
No.
If it was a homomorphism, then $f[(a+bi)(c+di)]=f(a+bi)f(c+di)=ac$
But $f[(a+bi)(c+di)]=f[(ac-bd) + (ad+bc)i]=ac-bd$
Verify the properties of a ring homomorphism $f$:
$f((a + ib)(c + id)) = f(ac + i(ad + bc) - bd) = ac - bd \neq f(a + ib)f(c + id) = ac$.
So the answer is no.
With even less computation: Since $i^2=-1$ the set $\{a+bi | a=0\}$ is not an ideal in $\mathbb Z[i]$