What is this pulley?
 

If you look at the link on the top left view of the engine (drive belts with #2),

http://foxed.ca/rx7manual/2003mazdar...002000W01.html

The strap going vertical is for the generator and the horizontal is for A/C.

I just wonder what does the pulley a the right on the A/C drive belt? Is it this one that drives the A/C or is it the second from the right one?

The middle part of the right one turns and the stops and then turns again when at idle. I want to know why it starts and stops rotating..

Thanks !

Keep reading that service manual ;)
The pulley on the right is for the A/C, the middle is the idle pulley.
If the pulley isn't spinning all the time, check belt tension or the bearing is seizing.

I said that the middle of the pulley is not turning all the time. The exterior of the pulley never stops turning with the strap. There is a clutch in this pulley but I want to know why it turns 15 seconds and stops 15 seconds and going on that way all the time. The middle isn't supposed to rotate if the A/C is off...I think...

And because of this when I'm on the idle at a red light the rpm is constant and when the pulley engages it makes the idle go down from 800 to 500 rpm and the car vibrates. It's annoying. It does that every 30 seconds.

I also figured out that there is always water on the ground under the car after a ride... This is the same water that the dehumidifier makes when the A/C is on...

It looks like the system is working even if the A/C is off but it doesn't produce cold air... And my A/C is working fine when I use it...

check the belt tension.

Read this: https://www.rx8club.com/series-i-trouble-shooting-95/ac-even-when-light-off-222022/

Thanks!