Arizona Law (A.R.S. § 28-812) specifically addresses this question. It is titled "Applicability of traffic laws to bicycle riders" and says that a person riding a bike on a road or road shoulder is granted all of the rights and is subject to all of the duties applicable to drivers of a vehicle.

That means that if a person on a bicycle violates any law which pertains to cars and traffic on a public roadway, they could not only be pulled over, they could be cited (given a ticket) and be punished (be fined, have their driver's license suspended, or even be arrested) under the traffic laws, just like a driver of a car could be.