Mikey -

I make sure the ring is a solid before proceeding.

I made a rectangle surface with an area equal to the width of the ring, times the circumference of the ring.

I positioned my extruded fingerprint so the desired depth extends below the rectangle.

Transform > Deform

Select Object to Flow: I select just the entire fingerprint.

Select Base Curve or Surface: I click on the rectangle on the right-hand side near the closest edge (shown by my red dot).

Select Target Surface Near Matching Side: Imagine the rectangle is wrapped around the ring. I click on the ring close to the corresponding place where I clicked on the rectangle (shown by my red dot).

After the fingerprint is flowed, I boolean diff the fingerprint which leaves the recessed areas.

Ed Ferguson