There are many places where one can find a guide on where to farm this ring, but nothing so far is certain. The guides mostly say where people have found the ring the most, but give no explanation as to why the ring might drop there as opposed to somewhere else. The aim of this thread will be the discussion and exploration of the drop system in order to determine where and why the ring drops.
For difficulties outside of Inferno, it seems that monster level dictates the item level of the items that the monster drops. For example, playing through Act II, one will notice drops of ilvl 14 and 16 rings early on, but as one progresses through the act (and into higher level monsters), one will notice those rings being replaced by ilvl 19 and eventually ilvl 21 rings. At first, it was thought that a monster could only drops items of equal or lower level to itself, but this was shown to be incorrect by the following video: http://www.youtube.com/watch?v=A9rv9Kh9_N8
It is not known how far item level can deviate from monster level, but it seems as if it deviates further down than up. One goal of this thread will be to determine just how far this deviation goes. On top of that, I would like to determine the approximate chances of getting items of one level relative to monster level to another. In other words, the chances of getting items of mlvl + 1, mlvl +0, mlvl -1, etc.
For this thread, we will assume that legendary items are like rares in that they need to roll off of a base item. In other words, the game first decides what type of item will drop, then decides if the item is legendary/rare/etc. The evidence for this is quite strong. For example, the legendary item The Crudest Boots, item level 6, can drop from monsters well above its level. I personally had the item drop from monsters at least level 15. If the game decided whether items would be legendary before determining a base item, it seems unlikely that it would pick an item so low. If this assumption turns out to be incorrect, we will find out.
Another goal is to find out if legendary items can roll up, down, or in both directions. For example, if items can only roll upwards, then Leoric's Signet will only be found in places with rings of item level 16 (and possibly 14). If it is only the opposite, then Leoric's Signet will only be found in places with rings of item level 19 (and possibly 21). In order to determine what directions legendaries can drop, a large sample of specific legendary drops needs to be compared with a large sample of item drops of the same type. For example, if we find get 30 Leoric's Signet drops in a place where the ring drops are a 70/30 split between ilvl 19 and 16, but only 15 in a place where ring drops are a 30/70 split between ilvl 19 and 16, it's a strong indication that legendaries can only roll down. Of course I just made all those numbers up, and it doesn't need to be Leoric's Signet necessarily, but since people are going to be farming it specifically, it's going to be the easiest item to collect data for.
How can you help? Accurate data collection. If you aren't sure of the source of an item drop, then the data is useless. The best method is learning the levels of the monsters you are fighting and slowly killing the monsters in small groups with single target abilities.
Credit: ztking for the video, Bloodseeker for his thread.