I am trying to prove this category has an initial object. However, I cannot seem to think of the initial object. I think once I know the object I can make the proof.
Let $N$ be a fixed rng (ring without identity) and $C$ be the category whose objects are rng homomorphisms $f: N \rightarrow R$ where $R$ is a ring (has identity). The morphisms of $C$ are the commutative diagrams
$\space \space N \space \space \space \space \space \space \space \space \space \space \space \space \space \space N$
$f$ $\downarrow$ $\space \space$ $\longrightarrow$ $\space \space \downarrow g$
$\space \space R$ $\space \space \space \space \space \space \space \space \space \space \space \space \space S$
I apologize for my diagram. Basically, denoting the horizontal arrow by $\alpha$, we have that $\alpha : R \rightarrow S$ is a ring homomorphism that makes the diamgram commute. So, $\alpha(f(n)) = g(n)$.
Could you please help me think of the initial object in this category.
You're looking for a universal ring $\tilde{N}$ living over the rng $N$. This would probably be something like the image of $N$ under an adjoint to the forgetful functor $\mathsf{Ring}\to\mathsf{Rng}$. Consider the ring $\tilde{N}$ which is $N\oplus\Bbb Z$ as an abelian group, with multiplication given by $$ (r,n)(s,m) := (rs + ns + mr, nm). $$
You can check that this ring does indeed have an identity ($1_{\tilde{N}} = (0,1)$), and there's a natural map $N\to \tilde{N}$ given by $n\mapsto (n,0)$, so this is a good candidate for initial object in this category.
You essentially need to check that a map $f : N\to R$ induces a map $\tilde{f} : \tilde{N}\to R$ making the relevant diagram commute (and that it is unique). Try the map $\tilde{f} : (r,n)\mapsto f(r) + n\cdot 1_R$ (it should be checked that this is actually a homomorphism).