Arizona is an at-will state and lawyers can fire for cause or for no cause but not if the cause violates the law. If an employer is going to terminate for a cause that violates the law, however, they will obviously state that the termination was without cause and you'll have the burden to prove otherwise.
The answer depends upon your employment status. If you are not under a contract with your employer or in a union, you can be fired for any reason or indeed for no reason (which would include a mistaken or incorrect reason). As the previous answer states, Arizona is an "at-will" state - that means you can quit without giving notice and you can be fired without notice. Of course, you cannot be fired for a constitutionally-protected reason (age , gender, national origin, for example) Generally, employees who work under an employment contract can only be terminated for reasons specified in the contract. If you are in a union, then your union representative will be able to inform you of your rights and will file a grievance on your behalf if they feel the termination violated the collective bargaining agreement. Arizona is a "Right to work" state meaning you cannot be fired merely becuase you refuse to join a union.
Many times, employers have an employee handbook which sets forth policies and procedures. Terminated employees often feel they have a claim against the employer if the employer did not follow the handbook's guidelines. But most handbooks and manuals state that nothign contained there chanegs the "at-will" status. The laws regarding an employer's duties and responsibilities arising under an employee handbook are complex, and a licensed attorney in your home state should be contacted to review individual circumstances.