Is it a stud or strapping

Can you send or post a picture of the situation? Studs would generally be in walls and not above a suspended ceiling. Hard to give good advice without a solid idea of what you have going on

Add a pic OP. Framing in the ceiling is generally called joists. The answer to your question is depends. Do you need to cut the full width of the joist to place the fixture? More importantly, why are you installing recess can lighting instead of flat led panels in 2023? The joist in question probably only supports the drywall hanging from it. If you can get away with cutting only the bottom half out it will probably be fine.

Attached picture to the post. I meant that they are joists, not studs.

Have you looked into those led recessed lights. They are only 0.5" thick and the ballast is on a 12" wire. These are so they can be installed anywhere even under ceiling joists.

Believe it or not, straight to jail.