From the perspective of a person, a rainbow is formed when the Sun is behind the person, and there is a critical angle made by the rainbow.
However, on several occasions, usually at noon when the Sun is higher, I saw a ring around the Sun made of the colors of the rainbow. Is that a rainbow? Is within the definition of a rainbow? And how is it possible?
What you're asking about sounds like an optical halo. It's produced by sunlight being refracted by ice crystals in the upper troposphere. The process is similar to that involved in a rainbow, except that the light is only refracted, not reflected, in this case.
As David mentioned above, it is not a rainbow, but an optical halo. Actually, halos are visible much more often than rainbows. Here you can find amazing pictures of halos (as well as rainbows and other optical phenomena) and some explanations of their appearance in the sky.
Well, the definition of a rainbow is "an arch of colors formed in the sky under certain circumstances" according to my Apple dictionary. More literally, it would need to be caused by rain, so you're correct that you'd see it 180° away from the sun.
But, if there are thin cirrus clouds made with tiny ice crystals, you can get what are called "sun dogs." Because of the angles of the faces of an ice crystal, sun dogs will form 22° away from the sun, not 180°. Wikipedia has some good pictures.
The other answers describe a rainbow-like phenomenon involving ice crystals, which may very well be what you saw. However, there is another possibility.
A normal rainbow occurs when light enters a spherical drop, refracts at the curved surface (dispersing the colors), reflects off the back of the drop, and then leaves the drop, refracting again. The angle between incident and outward-going light is about $42^\circ$, and so you see rainbows $42^\circ$ from the point directly opposite the Sun.
No one said, though, that the light had to undergo exactly one total internal reflection before leaving the drop. It can reflect multiple times, coming out at different angles. A second-order rainbow is at a slightly different angle from the normal first-order one. More interestingly, third- and fourth-order rainbows can be found circling the Sun (not circling the point opposite the Sun), simply due to the geometry. Wikipedia has some information, though unfortunately I cannot find any good diagrams for this effect online. Third-order rainbows are very hard to see, but they have been documented and are in a sense more "true" to the definition than phenomena involving ice.
