Suppose we have $\phi : \mathbb{Z}$ x $\mathbb{Z} \rightarrow \mathbb{Z}$ x $\mathbb{Z}$ , where $\phi$ is a ring homomorphism. Also, assume we know $\phi(1,0) = (a,b)$ and $\phi(0,1) = (c,d)$. Then suppose we want to know the mapping of some other $(m,n) \in \mathbb{Z}$ x $\mathbb{Z}$.
In general, will $\phi(m,n) = m\cdot(a,b) + n\cdot(c,d)$ ?
My reasoning here is the following:
$\phi(m,n) = \phi(m\cdot(1,0)+n\cdot(0,1))$. Then, since $\phi$ is a ring homomorphism, this is equal to $\phi(m\cdot(1,0)) + \phi(n\cdot(0,1))$, and since $m$ and $n$ and simply constants, this equals $m\cdot\phi(1,0) + n\cdot\phi(0,1)$ = $m\cdot(a,b) + n\cdot(c,d)$.
Does this logic check out? Any help would be greatly appreciated.