Prey’s "With This Ring …" optional objective begins when you reach the Cargo Bay during "Shipping and Receiving" and you meet the other survivors. One of the survivors on the ground floor — in the area where you set up the turrets as part of "Cargo Bay Defenses" — is Kevin Hague. He’ll ask you if you’ve seen his wife.
Look for Nicole Hague
You can pull up Nicole Hague’s tracking bracelet on any security station computer, or you can read the next sentence. She’s 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 it in the supply closet on the second floor to the left of the Yellow Tulip. (This is the same closet where you investigate Captain Stabfellow’s map in "Treasure Hunt.")
Recover the wedding ring
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. Shoot her and loot her corpse for a neuromod, a recycler charge and the wedding ring.
Return the Wedding Ring to Kevin Hague
Now you just have to return to Kevin Hague in the Cargo Bay with the sad news to complete the "With This Ring …" optional objective.