Amulets of life saving only generate when a lich generates, either in its
inventory or somewhere else on the level where it first generated. Each amulet
is tied to its lich, and the lich can only be permanently killed when its amulet
is destroyed (including by the player using it up to save their own life, but
not limited to that; any method of destroying it such as looting it from a
cursed bag of holding would work). If the amulet remains intact, the lich will
eventually resurrect.
If a lich is genocided, any amulets connected to liches of that type aren’t
destroyed, but stop being effective to save lives. Possibly there should be some
visible indicator of this, such as the amulet being cracked. If all liches are
genocided, then no amulets of life saving will work.
Not specified what would happen if an amulet of life saving is wished for;
probably either the amulet doesn’t work, or it spawns a random lich it’s
connected to somewhere on the level.