Chances are good its a stainless steel. The vast majority of stainless steel alloys have little to no magnetic attraction. This was reinforced in my mind when I tried to build a magnetically attached stainless steel platform for my 3D printer and - spoiler alert - it didn't stick. The early stainless steel fridges, and even many on the market today, exhibit little desire to hold fridge magnets, so you have to use strong magnets to hold small sheets of paper. Today a lot of "stainless steel" fridges are stainless clad carbon steel, both to reduce cost and to stop consumer complaints about lack of magnetic attraction.
The cheapest steel - carbon steel - is magnetically attractive, but is not hypoallergenic, and is prone to rusting against human skin, so is less likely to be used in rings and jewelry. Surgical stainless steel seems to be a common name for the alloy used in steel jewelry, when the reality is that it often contains nickel, a metal which does raise an allergic reaction in a sizable portion of the population. They're still allowed to call it "hypoallergenic" which, if anything, shows the meaninglessness of that term in relation to jewelry. My understanding is that it's much more strict when used for cosmetics.
The most dangerous objects are large unsecured pieces of metal, such as oxygen tanks, wheelchairs, cleaning equipment, and IV poles. Small pieces of metal, such as rings, can cause bruising and minor injury, but shouldn't result in the kind of catastrophic injuries MRIs are known for. The more dangerous "small piece of metal" incidents are usually aneurysm clips and pacemakers, though there is one report of injury due to a
hair pin the patient had left in their hair - it apparently lodged in their pharynx through their nose and they had to undergo a procedure to remove it.
I expect that even if your ring was very magnetically attractive carbon steel you would still be able to manipulate it within the MRI by hand with some effort. It would only become marginally dangerous if it left your finger, and danced around the MRI opening - and even then it's unlikely you'd suffer more than minor bruising and perhaps small lacerations.