I was thinking of the stainless steel version, but looking that up looks like the answer is "depends". Depends on the definition of "magnetic" (since all materials are to a degree) and the type of stainless steel.

I just noticed that my own "iron" ring (which is stainless steel) doesn't seem to respond to typical magnets you'd find around the office.

Clearly _iron_ is though, considering it's the element with the symbol "Fe" which is short for Ferrum.

Usually (but not always) higher quality stainless steel is not magnetic, while the lower quality is slightly magnetic.

However even slightly magnetic can be a problem in an MRI - and worse, there is no practical way for the MRI tech to tell how the stainless steel will act, so they will err on the side of caution.