If $R$ is a ring with unit element $1$ and $φ$

Question

If $R$ is a ring with unit element $1$ and $φ$ is a homomorphism of $R$ into an integral domain $R'$ such that kernel of $φ$ is different from $R$, prove that φ(1) is the unit element of $R'$.

---

could you give me tips to solve this?

Answer 1

$\phi(1)=b$ is not zero, since the image is not zero. We have $\phi(1\cdot 1)=\phi(1)\cdot\phi(1)$ implies that $b=b\cdot b$ and $b(1-b)=0$. Since $R'$ is an integral domain and $b\neq 0$, $b=1$.