When is the ring homomorphism $\mathbb{Z} \to R$ an epimorphism? For every ring $R$ there exists a unique ring homomorphism $f: \mathbb{Z} \to R$.  For which rings is this an epimorphism?
The only examples I know are the subrings of $\mathbb{Q}$ and the quotient rings of $\mathbb{Z}$, namely the rings $\mathbb{Z}/\langle n \rangle$.  Are these all, or are there more?
Three characterizations of epimorphisms of commutative rings are listed here:

*

*Epimorphisms of rings, The Stacks Project.

Maybe one will help.
(Why I'm interested: an object in a category is subterminal if its unique morphism to the terminal object is a monomorphism.  Since $\mathbb{Z}$ is initial in $\mathrm{Ring}$, here I am asking what are the subterminal objects in $\mathrm{Ring}^{\rm op}$.)
 A: Let me summarize what I've learned from the comments.  A ring $R$ for which the unique homomorphism from $\mathbb{Z}$ to $R$ is an epimorphism is called a solid ring.  Such rings are necessarily commutative, by Prop. 1.3 (b) of this paper:

*

*H. H. Storrer, Epimorphic extensions of non-commutative rings, Commentarii Mathematici Helvetici 48 (1973), 72–86.

By well-known results in commutative ring theory, a solid ring is thus the same as any of these:

*

*a commutative ring $R$ for which the multiplication map $m \colon R \otimes R \to R$ is an isomorphism;


*a commutative ring $R$ such that the forgetful functor $R\, \mathsf{Mod} \to \mathsf{AbGp}$ is full.


*a commutative ring whose core $ cR = \{r \in R: r \otimes 1 = 1 \otimes r \in R \otimes_\mathbb{Z} R \} $ is all of $R$.
Commutative solid rings, and thus all solid rings, were classified here:

*

*A. K. Bousfield and D. M. Kan, The core of a ring, Journal of Pure and Applied Algebra 2 (1972), 73–81.

The following rings are solid:

*

*$\mathbb{Z}/n$ for any $n$.

*any subring $R \subseteq \mathbb{Q}$.   Such a subring is always of the form $\mathbb{Z}[P^{-1}]$, meaning the ring of fractions whose denominators (in lowest terms) are divisible only by the primes in some set $P$ of primes.

*any ring of the form $\mathbb{Z}[P^{-1}] \times \mathbb{Z}/n$ where each prime factor of $n$ is in $P$.

Bousfeld and Kan show that in the category of commutative rings, every colimit of solid rings is solid.   They also show that every solid ring is a colimit, in the category of commutative rings, of solid rings of the above three types.
Bousfeld and Kan also give a more explicit description of all the solid rings.  They show that every solid ring is either of types 1-3 or of a fourth type:


*$c(\mathbb{Z}[P^{-1}] \times \prod_{p \in Q} \mathbb{Z}/p^{e(p)}) $, where $P$ and $Q$ are infinite sets of primes with $Q \subseteq P$ and each $e(p)$ is a positive integer.   (Here $c$ stands for the core, as defined above.)

