Unnatural homomorphism form domain $R$ to $Frac (R)$

Question

There is a natural homomorphism for $R$ to $Frac (R)$ that sends $r\rightarrow(r,1)$, but beside this injective homomorphism, is there example of ring $R$ s.t there exist another injective homomorphism (unnatural homomorphism!) from $R$ to $Frac (R)$?

Answer 1

Every time the field $Frac(R)$ has an endomorphism you can compose it with the usual map $R\to Frac(R)$ to get a new embedding. As fields tend to have many endomorphisms, this gives you lots of examples.

In fact, thi gives you all examples. If $f:R\to Frac(R)$ is a morphism, it maps non-zero elements to invertible elements, the univrsal property of the field of quotients implies that there exists a ring homomorphism $\bar f:Frac(R)\to Frac(R)$ such that $f=\bar f\circ i$, with $i:R\to Frac(R)$ the normal inclusion (and $f$ is the usual map if and only if $\bar f$ is the identity, since $i$ is an epimorphism of rings)

Answer 2

If $R$ is a field itself, then $R \cong \text{Frac}(R)$, so we just have to find a non-trivial embedding of $R$ into itself. We now have many options, e.g. $R =\mathbb{C}$ and the map defined by complex conjugation.