ag.algebraic geometry - Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective

I would like to know under what condition the morphism $mathcal{O}_Ylongrightarrow f_ast mathcal{O}_X$ induced by a morphism $f:Xlongrightarrow Y$ of schemes is injective.

Let me give an example (which I'm not completely sure about though).

I believe, if $X$ and $Y$ are reduced and $f$ is surjective and closed, the morphism $mathcal{O}_Y longrightarrow f_ast mathcal{O}_X$ is injective.

(Thus, proper flat morphisms of varieties have this property.)

Maybe one could forget about schemes and give a condition for locally ringed spaces?