Cleanup (Issue: Two versions)
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Dragonsreach Dungeon, in a an iron pot in the tunnel directly below one of the cells. The tunnel is accessed through a loose grate with a dead body lying on top of it. The loose grate is in the cell you end up in when you are taken to jail.
With Cutting Room Floor (the SSE version, from v3.0.7 and onwards) and its Legacy patch installed, the Ring will instead appear in Nightingale Hall once the Thieves Guild quest is over (this information is not shown in the mod page, but is shown in the activity logs and the txt file included with the mod.)
There's a note in the pot with locations of some other rings - Warlock, Vampiric, Mentor, Phynaster.