Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \to Spec(R)$ to be injective as a map of topological spaces (completely disregarding any of our sheaf structure)? That is, every prime that lifts lifts to a unique prime? If this problem is intractable asis, I can add more conditions, so please don't add answers unless they are actual answers.

Rather than listing various nontrivial sufficient conditions, let me give you a reference. Search EGA for the word "radiciel" (or start reading at Definition 3.5.4 of EGA I). A morphism of schemes $X\to Y$ is said to be radiciel (or universally injective) if for every field $K$, the induced map on $K$points $X(K)\to Y(K)$ is injective. This generalizes the notion of a purely inseparable field extension. 


If $S/R$ is purely inseparable then the primes will lift uniquely. 

