Are there many examples of epimorphisms in the category of unital rings from a commutative ring to a noncommutative division ring? I apologise for the question. I am not very familiar with ring theory and could not find an answer to this question.
Here is why I am asking this question. Let $A$ be a unital ring. There is a result due to Cohn and Malcolmson which says that ring epimorphisms from $A$ to a division ring (up to post-composition with an isomorphism) are in bijection with integer-valued Sylvester rank functions on $A$ (see Theorem 7.5 in A. H. Schofield, "Representation of rings over skew fields"). I recall that a Sylvester rank function $\rho$ assigns to a finitely presented $A$-module a non-negative real number and satisfies the following axioms:
(1) $\rho(X\oplus Y)=\rho(X)+\rho(Y)$ for every finitely presented $A$-modules $X$ and $Y$;
(2) $\rho(Z)\leq \rho (Y)\leq \rho(X) +\rho(Z)$ for every right exact sequence of finitely presented $A$-modules $X\to Y\to Z \to 0$;
(3) $\rho(A)=1$.
When $A$ is a commutative ring, the ring epimorphisms from $A$ to a field are parametrised (up to isomorphism) by the prime ideals of $A$. Each prime ideal $p$ of $A$ gives rise to a distinct ring epimorphism via the composition $A \to A_p \to k (p)$ from $A$ to its localisation at $p$ and then to the residue field of $p$. I was wondering if there are many more epimorphisms from a commutative ring $A$ to a division ring and therefore also corresponding integer-valued Sylvester rank functions.