There is nothing that you can do to get rid of mushrooms. Although they are a fungi fungicides do nothing to mushrooms. This would be about the same thing as trying to kill a giant oak
tree with a shot of dandelion killer.
They come from spores which are trapped usually in pieces of buried wood. as the wood decays the spores are released and then work their way to the surface. Actually the mushroom itself is just the
flower of the fungi, the actual infected area came be much larger. The rings are known as Fairy Rings. Most of the time, they are harmless to the lawn, however if the mycelium growth, (the actual fungal body which appears as very fine strands which resemble spider webs)gets to thick, it can choke out the lawn. The best thing to do if that is the case is core aerate the lawn DO NOT USE A SPIKER! as this will compact the area and make the problem worse.
Fairy rings have a lot of really cool lore behind them, and are pretty interesting to read up on. They can be anywhere from a few feet to over a mile across. They have been observed in the rain forests by the astronauts circling the earth in outer space!
Here are some interesting links:
http://www.ext.colostate.edu/pubs/garden/02908.html http://www.unce.unr.edu/publications/FS03/FS0320.pdf#search=\'fairy%20ring%20fact%20sheet%200320\'