Proposition 1.2.
Let $A$ be a ring $\neq 0$. Then the following are equivalent:
i) $A$ is a field
ii) the only ideals in $A$ are $0$ and $(1)$
iii) every homomorphism of $A$ into a non-zero ring $B$ is injective.
I am confused by how ii) implies iii). Given a homomorphism of $A$ into a non-zero ring $B$, the kernel of the homomorphism is an ideal. So by ii), it has to be $0$ or $(1)$. If it is $0$, $A$ is injective. But why can't it be $(1)$?