For an exercise, I need to show that
$$\phi: \mathbb{Z_n} \rightarrow \mathbb{Z_{n_1}} \times \mathbb{Z_{n_2}}; x \mapsto (x\mod n_1, x\mod n_2)$$ where $n = n_1 \cdot n_2$ and $n_1, n_2$ coprime is a Ring Homomorphism.
As far as I understand, this means I need to show that: $\phi(a + b) = \phi(a) + \phi(b)$ and $\phi(a \cdot b) = \phi(a) \times \phi(b)$ and $\phi(1_{Z_n}) = (1_{Z_{n_1}},1_{Z_{n_2}})$.
My approach so far: $\phi(a +b) = (a + b \mod n_1, a + b \mod n_2)$ and $\phi(a) + \phi(b) = (a \mod n_1, a \mod n_2) + (b \mod n_1, b \mod n_2)$.
Here I am stuck since it is not clear to me how $+$ and $\cdot$ are defined in $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2}$. If they are defined as $(a \mod n_1, a \mod n_2) + (b \mod n_1, b \mod n_2) = (a + b \mod n_1, a +b \mod n_2)$, then wouldn't that make the exercise trivial?