When the Americans reached the moon, why was the flag flying, while there were no air in the moon?

The flag was not flying. It only looks that way in still photographs. The flag had a wire sewn into it's top edge, which held the flag outward from the pole. Look at videos (below) and you will see that the flag is stationary except when an astronaut bumps the pole.

The flag wasn't exactly flying...

The astronauts actually put in bent wire to make the flag appear to be flying, when it is really not. And it also was moving when the astronauts were putting it in place. I also just thought of this theory that even though there is no air on the moon, when the lander actually landed, it could have pushed the air around, like if you jump into a pool, the water moves outward. So, instead of air or water, we have moon dust. And it also was moving when the astronauts were putting it in place.
Hope this worked :P
~Nick

It's hung from a horizontal rod

You're looking at a still photograph. There is no way you can say, from looking at ONE still photograph, that the flag was moving. In fact, if you look at other still photographs taken of the same flag at different times and from different angles, you will see the exact same "waves" in the flag in each one. And, THAT, my friend, definitively proves that the flag was NOT moving.
http://www.collectspace.com/review/apollo11_flag01.jpg
http://www.hq.nasa.gov/alsj/a11/a11det5875VertCrop.jpg
http://img.timeinc.net/time/2007/sputnik_anniv/apollo_11.jpg

There was a rod sewn into the top of the flag to keep it from hanging limply off the pole. The wrinkles in the flag are from being folded on the outward trip.

The picture you used in your question is the famous photo of Buzz Aldrin saluting the flag. You can see part of his hand near his visor. Here is a higher resolution of the image:

http://www.hq.nasa.gov/office/pao/History/alsj/a11/AS11-40-5874.jpg

If you look carefully at the top of the flag you can see how the fabric of the flag wraps around the rod.

Now, take a look at the picture taken moments later. Aldrin is still standing on the same spot but he has lowered his hand and turned his torso slightly to his left:

https://www.flickr.com/photos/gsfc/4812876306

Note that the flag looks exactly the same in both pictures, something that would be impossible if it was flapping in the wind.

Because the pole of the flag was vibrating.

Wires were inserted inside the flag to hold the flag up.

It isn't the flag was stiff. BECAUSE THEY MADE IT THAT WAY AS THERE IS NO AIR!

There's a wire through the top seam of the flag, holding it out.

That flag was hanging on a pole with a cross bar at the top, to hold it "out". This has been answered many times before.

Actually, the fact that the flag kept moving for while after being put up is proof that there was no air - if you try making a flag as described in the other answers and put it up in your living room and wiggle it, it will stop moving almost instantly due to air drag. Only if it is in a vacuum will the cloth keep moving for any time. Of course the wrinkles in the still picture are due to it being packed for the trip.

It wiggled and jiggled ONLY when the astronaut's hand was on it. Soon's he let go, it stopped.

because there is very less gravity on moon, so the flag is in state of inertia for a time.

I did not see a definite answer, each one speaks the words of various

But is not it possible that because of the low gravity of the moon that flies the flag

It was starcched as they knew there was no wind to furl it.

The flag was deliberately designed that way.

the flag was not flying