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.