I have been reading through Larry Niven's The Integral Trees, which concerns a gas ring around a neutron star with a habitable center that humans became stranded on., The atmosphere of which was stripped from a nearby gas giant. I was wondering if such a gas ring could exist around a world that is just like earth aside from there being no moon. How could this come about? I was thinking a series of ice comet impacts could have blown part of the atmosphere away from the planet where it became locked into orbit, meanwhile the planet's atmosphere was replenished due to the water from the comets. This does seem unlikely but thought I would ask.
Mr. Niven's book allowed us to suspend our disbelief because of the gravitational pull of the neutron star. An horrendous amount of matter could be held in orbit around the star — enough to permit the idea of a habitable jelly-filled donut with the star at the center.
Regrettably, you have two really big cons for a planet the size of earth to have such a thing.
We're probably not be big enough. A simple gas within the ring would simply bubble off into space no matter how much there was. Give the ring enough mass to hold the atmosphere together (you know, like a jelly donut...) and you might have enough competing gravity to simply yank the ring to the ground. That's why the neutron star was such a brilliant idea. There was so much gravity in play the ring was believable.
The bigger problem, however, was one Mr. Niven didn't need to worry about: the Van Allen Radiation Belts (why? Because (a) a neutron star's magnetosphere is massive and (b) a neutron star isn't spitting out solar wind like our boring G-type star). That protective aspect of our magnetosphere permits life on Earth at all. Those belts are a whole lot closer to the Earth than the moon, and a whole lot closer than would permit a habitable ring system. This is where I just ache to say it, but without those rings, you can have all the atmosphere you want and the ring still won't be habitable.
HOWEVER Please do not give up on the idea! I believe you can overcome the problems with some adjustments to your planet.
Make your planet larger than earth. Indeed, make it as large as astonomers believe a habitable world can be. The larger the world, the more substantial the ring system, and the larger the radiation belts.
Modify the core such that a stronger magnetic field is generated. Make it more dense, or a higher percentage of iron, or inject the core with an unusually high oxygen component to induce greater ferrousnessness... ness... Or have the core spin faster (which would even be believable with a larger world). Anything to justify a larger magnetic field.
Finally, you might consider really dense "rocks" as part of your ring. Something to exert a bit o' gravity to hold the ring together and keep the atmosphere from bubbling off into space. Make the rocks magnetically repellant so they create a shield around the jelly donut with the atmosphere at the center.
And if none of that works, consider a habitable moon around a gas giant sporting a habitable ring between the planet and the moon.
Nope. Gas torii are real things (there's one around Jupiter created by Io, and Enceladus produces one around Saturn), but they are wispy things. They are the result of gas being ejected or stripped from an orbiting body, so right away we can say that no such thing could exist around an Earth with no moon, because you need the moon to create the ring! Stripping gas from a moon produces such a ring because the individual gas particles remain in orbit around the primary, pretty close to the orbit they started in--that being the orbit of the moon itself!--with the relatively minor perturbations caused by whatever stripped them from the moon causing them to spread out into a torus.
There is, however, no restoring force holding those particles together as a coherent ring against internal pressure. They persist only due to a combination of being so thin that any individual gas particle is highly likely to complete a whole orbit without actually bumping into any other gas particle, and being continually replenished by the source moon. If a ring were thick enough to have noticeable pressure in the middle, it would simply blow itself apart.
Note that Niven's Smoke Ring is fed by the continual loss of gas from Goldblatt's World, which is stripped off by Voy's (the neutron star's) tides. It takes a huge continuous input of new gas to maintain it. Even so, the Smoke Ring is not particularly plausible itself. Niven gets away with it because the story is good, the Rule of Cool is in operation, and the environment is exotic enough that the average reader can't immediately recognize it as implausible. After all, who knows what can happen in the extreme environment around a neutron star stripping gas from a close gas giant? In reality, what should happen is that you get a hot, violent accretion disk, which spirals inwards due to loss of energy to internal friction from tidal shear--not a cozy, habitable Smoke Ring at all! Try to put it around an Earthlike planet instead, and familiarity kills your suspension of disbelief.
You could, however, go with an artificial gas torus, held in by a thin, solid, artificial membrane, with active stationkeeping. That's the approach taken by the Orion's Arm worldbuilding project, for example.
Small gas cloud of near atmospheric pressure in vacuum dissipates almost instantly. Why Niven's torus does not? Even though it is huge, it's self-gravitation is too weak to hold it together - gas has low density. It is held together by tidal forces from neutron star.
A planet would be several orders of magnitude lighter than star, and it's tidal forces would be accordingly smaller. So it would be able to hold torus with 1/1000 pressure of Niven's torus at best.
A gas ring is possible, but would be most likely in a highly reactive environment. Perhaps caused by high levels of meteoritic bombardment, extreme volcanism or similar. A major problem would be reactive elements such as oxygen reacting with other elements in the vicinity especially hydrogen. So yes to the gas ring but no to an oxygen rich gas ring, unless you can invent some form of special pleading.