Ownership of a PhD Student's Research
If your advisor hasn’t contributed anything to the paper intellectually then they can’t ethically put their name on a paper they didn’t contribute to. That has nothing to do with funding or RAships.

That said, what I described above is the ideal. Students are often under pressure to do what advisors say. Authorship is best discussed directly and if you have any issues you should speak with your advisor and establish standards in your group.