So I am taking a look at ring homomorphisms, and two examples in the book which are not stated that they are ring homomorphisms, but I think they are not Ring homomorphisms, am I correct?
$\text{i})$ $\phi : \mathbb {C} \rightarrow \mathbb {C}$, given by $\phi(x)=-x$
$\text{ii})$$\phi : \mathbb {C} \rightarrow \mathbb {C}$, given by $\phi(x)=x^2$
On
To be a ring homomorphism (with 1) it has to hold, that
1) $\phi(x+y)=\phi(x)+\phi(y)$
2) $\phi(x\cdot y)=\phi(x)\cdot \phi(y)$
3) $\phi(1)=1$
For your first example it is $\phi(1)=-1\neq 1$. Hence not a ring homomorphism. For your second example it is $\phi(x+y)=(x+y)^2=x^2+2xy+y^2\neq x^2+y^2=\phi(x)+\phi(y)$
Hint: For (i), where does a ring homomorphism have to map $1$? For (ii), is $\phi(x+y) = \phi(x) + \phi(y)$?