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

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.

• So what are the morphisms here? What is a morphism from $(R, r)$ to $(S, s)$ in this category? Is it merely a ring homomorphism from $R$ to $S$? (No; why?) – M. Vinay Mar 20 '19 at 7:40
• Okay, you answered that part in your updated question. That leads to the answer. – M. Vinay Mar 20 '19 at 7:41
• At least intuitively, it seems to make more sense that it would be the ring formed by the integers with the usual multiplication/etc., with any integer as the fixed point. After all, we can't have a ring with no elements, and the initial object in the category of rings is the usual ring of the integers, so I'm not exactly sure how there would be a difference if you fix a particular point. (This as opposed to the category of sets - with initial object the empty set - and the category of pointed sets - with initial object a singleton set.) Maybe I'm overlooking something obvious though. – Eevee Trainer Mar 20 '19 at 7:41
• Homomorphism can't just change one element to another. It must save structure unlike pointed sets – Ivan Sharapenkov Mar 20 '19 at 7:53
• I would presume it is $(\mathbb{Z}\langle x \rangle , x)$ given that the property he wants is more or less the universal property of the free algebra over $\mathbb{Z}$. (every ring with $1$ has a canonical map from $\mathbb{Z}$ into it and the marked point gives the evaluation of $x$, so the univ property of the free algebra gives us a unique morhism from $\mathbb{Z}\langle x \rangle$ into any pointed ring sending $x$ to $r$) – Enkidu Mar 20 '19 at 9:10

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.