Chapter 6.2, Problem 10E

Let $\theta :{\mathbb{Z}}_3\to {\mathbb{Z}}_{12}$ be defined by $\theta \left({\left[x\right]}_3\right)=4{\left[x\right]}_{12}$ using the same notational convention as in Exercise 9.

Prove that $\theta$ is a ring homomorphism.

Is $\theta \left(e\right)=e^{\text{'}}$ where $e$ is the unity in ${\mathbb{Z}}_3$ and $e^{\text{'}}$ is the unity in ${\mathbb{Z}}_{12}$?