How to find initial object of the category of pointed rings?

Question

I have the category of pointed rings. Objects are all pairs $(R, r)$ where $R$ - ring (with 1) and r is the element of R. Morphisms are homomorphism of rings. Morphism $(R, r) \longrightarrow (R', r')$ exist if exist homomorphism p: $R \longrightarrow R'$ and $p(r)=r'$.

Can somebody help to find initial object or proof that it doesn't exist? I think that it doesn't exist but can't find a way to proof.

Thanks.

Answer 1

It is up to unique isomorphism $(\mathbb{Z}[x],x)$ as this has the universal property that any assignement $x \mapsto r\in R$ gives a unique morphism $$\mathbb{Z}[x] \to R$$ hence any choice of $(r,R)$ gives you such a unique map and hence it is the initial object.