Ceribia gives you a very good list of reasons why a full fledged invisibility ring is a monster item. By default, there are no limits to the number of time you can trigger the magic and become invisible. This means, even though it is only a level 2 spell, it is as if you had a wizard with a 14,100 slots of level 2... (60 * 60 * 24 = 86400 seconds per day, divided by 6 [turn duration] is 14,100 -- that's assuming you bypass sleeping, though, and you are constantly activating the ring...)
If you are the DM and want to offer such an item of lower greatness, you can do so by offering a ring of invisibility that only works 1 to 3 times a day. Maybe it has a curse (like The One Ring) and maybe the player does not know how many charges are available on that day. You could also limit it in time (i.e. the invisibility could last just 1 minute like the spell.)
You mentioned the ring of regeneration. I actually gave such a ring to a 1st level player, only that ring (1) only started kicking in if you reached 0 hit point, (2) only worked after 1d4 rounds [so you could still die before it worked], (3) only gave 1 hit point every hour after that [so max. 24 in a day and sleeping a whole night anyway gives you all your hit points!], (4) it was cursed, if you removed it, you'd be killed by the curse after about 4 hours; the curse was to make you tired, as if you were hit by one level of exhaustion every hour or so. The player lost it to a deck of card, though.