Given a ring $R$, a prime ideal $mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $Rto S$ 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 as-is, I can add more conditions, so please don't add answers unless they are actual answers.
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