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Suppose to have $\phi$ a ring morphsim from $A$ to $B$, let $X=SpecA$ , $Y=SpecB$ and $\psi$ the induced morphism of affine schemes. It's true that if $\psi$ dominant than $\phi$ is injective?

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No, but yes for reduced rings. It may help to do exercise II.3.11 in Hartshorne.

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