Is it permissible to sacrifice one's own life to save another? There are many noble examples that come to mind, such as donating your heart to save a child or throwing yourself on a grenade. Is this considered shvichat damim, and thus in the category of sheiv v'al ta'aseh?
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In wartime it's a different matter.
In peacetime, one who puts themselves in "severe danger" to save a life is termed a "chasid shoteh", a "super-righteous lunatic."
Need to add more, including sources. Watch this space.
It is forbidden to commit suicide, even for a noble purpose. When it comes to saving another Jew's life at the expense of one's own life, the halachah tells us that there is no way for humans to calculate whose blood is more precious in G-d's eyes. Therefore, one must always follow the passive path if one person will certainly die. If it is possible that both people will be saved, the question is more complex: the poskim discuss if it is permissible to put oneself in danger to save another person from certain death. A lot depends on the relevant practical conditions, and one must consult a knowledgeable posek.