I want to make a D&D criminal sorcerer, whose story would be that he scammed a ring off a novice mage, and that ring then gave him his magical powers. Would it be valid to use this ring as an arcane focus? Or would that be OP as I would not risk dropping it? The rulebook says one can use a crystal as a focus, so could I put the crystal inside the ring?
The main game-mechanical reason why spellcasting foci (or component pouches, their alternative) exist is that they offer a way to disarm a spellcaster by taking them away. Disarming the PCs can be used by the DM as a tool to get the party into a situation they can't solve through combat. This opens up opportunities to apply non-combat skills and roleplaying.
Many of the more munchkinesque attempts to loosely interpret what a focus can and can't be try to somehow circumvent such situations. But a ring can still be taken away just like most foci. When the ring is a very plain one, there is a good chance that one might miss it when frisking your character. But you could compensate for that by making the ring way more flashy, maybe visibly glowing and pulsating with magic power so even the dumbest NPC recognizes it as important.
But I am not your DM. Ask your DM if they allow it or not.