How many homomorphisms are there of $\Bbb Z \times \Bbb Z\times \Bbb Z$ into $\Bbb Z$? I've tried looking around for an explanation to this problem, but I've been having trouble finding a clear solution that specifically focuses on this question:
How many homomorphisms are there of $\Bbb Z \times \Bbb Z\times \Bbb Z$ into $\Bbb Z$?
I'm studying abstract algebra and we have just begun discussing rings. Thank you in advance for your help. I would appreciate it if you could give me step by step help for this problem. This is not for homework- I just need to figure out how to do this problem to prepare for my exam. Thanks!
Edit: Here is a post that addressed the question, but did not answer it fully: Describe all ring homomorphisms

Assuming the question is about ring homomorphisms, there are $3$. Here is how to see this very explicitly. 
Write $e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1)$. These three elements form a complete set of orthogonal idempotents. This means that they satisfy $e_i e_j = \delta_{ij} e_j$ and $\sum e_i = 1$, or more explicitly
$$e_i^2 = e_i, e_i e_j = 0 \text{ if } i \neq j, e_1 + e_2 + e_3 = 1.$$
The significance of these conditions for us is that they must be preserved by ring homomorphisms. Hence if $f : \mathbb{Z}^3 \to \mathbb{Z}$ is a ring homomorphism, $f(e_i)$ is a complete set of orthogonal idempotents in $\mathbb{Z}$. It's clear that $f$ is determined by the $f(e_i)$.
But $\mathbb{Z}$ only has two idempotents, namely $0$ and $1$. The condition that $\sum f(e_i) = 1$ means exactly one of the $f(e_i)$ is equal to $1$, and then orthogonality is automatic. Hence there are $3$ homomorphisms corresponding to whether $f(e_1), f(e_2)$, or $f(e_3)$ is $1$; these are the $3$ projection maps $\mathbb{Z}^3 \to \mathbb{Z}$.
More generally, homomorphisms $\mathbb{Z}^n \to \mathbb{Z}^m$ correspond to functions $m \to n$. 
The algebraic significance of complete sets of orthogonal idempotents in a commutative ring $R$ is that they correspond to decompositions of $R$ as a finite product $\prod_i R e_i$ of rings. The geometric significance is that they correspond to decompositions of $\text{Spec } R$ into a finite disjoint union $\coprod_i \text{Spec } R e_i$ of affine schemes. Here $\text{Spec } \mathbb{Z}^n$ decomposes into $n$ copies of $\text{Spec } \mathbb{Z}$. 
