Morphisms in the category of rings We know that in the category of (unitary) rings, $\mathbb{Z}$ is the itinial object, i.e. it is the only ring such that for each ring $A,$ there exists a unique ring homomorphism $f:\mathbb{Z} \to A$.
This means, in particular, that $\mathbb{R}$ does not satisfy this property, so for a certain ring $B$, we can construct $g:\mathbb{R} \to B$ and $h:\mathbb{R} \to B$ such that $g \ne h$.
So far, I have proven that $B$ cannot be an ordered ring and cannot be $\mathbb{R}^n \ (n\in\mathbb{N})$. Can you help me finding such a ring $B$?

$\mathbb{R}$ has two distinct embeddings into the tensor product $\mathbb{R}\otimes_{\mathbb{Z}} \mathbb{R}$, given by $r\mapsto r\otimes 1$ and $r\mapsto 1\otimes r$. 
These are the two inclusions coming from the fact that the tensor product is the coproduct in the category of rings.
