With This Ring... is a side quest in Prey (2017).
Nicole Hague can be found in the Executive Suites area of the Crew Quarters. You’ll need the Executive Suites grav lift keycode to get up there; you can find this in the supply closet on the second floor, to the left of the Yellow Tulip.
The first time you enter the Executive Suites, you’ll encounter Nicole in her phantom form. She’s in the Guest Suite at the end of the hall along with the phantom of Argenteno Pero. Kill her and loot her corpse for a neuromod, a recycler charge and her wedding ring, which bears an inscription from her husband, Kevin Hague.
Return the ring to Kevin who will be vaguely upset to learn what happened to his wife. At this point the quest will complete.