You are correct... and Rowbear was incorrect. A criminal background check will show an arrest and will also show convictions. And it will last forever, unless expunged (or set-aside).
There are statutes on setting-aside a record. The first applies to only arrests with no conviction and the second applies to convictions.
Arrest only:
ARS §13-4051:
"Entry on records; stipulation; court order
A. Any person who is wrongfully arrested, indicted or otherwise charged for any crime may petition the superior court for entry upon all court records, police records and any other records of any other agency relating to such arrest or indictment a notation that the person has been cleared.
B. After a hearing on the petition, if the judge believes that justice will be served by such entry, the judge shall issue the order requiring the entry that the person has been cleared on such records, with accompanying justification therefor, and shall cause a copy of such order to be delivered to all law enforcement agencies and courts. The order shall further require that all law enforcement agencies and courts shall not release copies of such records to any person except upon order of the court.
C. Any person who has notice of such order and fails to comply with the court order issued pursuant to this section shall be liable to the person for damages from such failure."
Source:
http://www.azleg.state.az.us/ars/13/04051.htm
Conviction:
ARS §13-907:
"Setting aside judgment of convicted person on discharge; making of application; release from disabilities; exceptions.
A. Except as provided in Subsection B of this section, every person convicted of a criminal offense may, upon fulfillment of the conditions of probation or sentence and discharge by the court, apply to the judge, justice of the peace or magistrate who pronounced sentence or imposed probation or such judge, justice of the peace or magistrate's successor in office to have the judgment of guilt set aside. The convicted person shall be informed of this right at the time of discharge. The application to set aside the judgment may be made by the convicted person or by the convicted person's attorney or probation officer authorized in writing. If the judge, justice of the peace or magistrate grants the application, the judge, justice of the peace or magistrate shall set aside the judgment of guilt, dismiss the accusations or information and order that the person be released from all penalties and disabilities resulting from the conviction other than those imposed by the department of transportation pursuant to section 28-3304, 28-3306, 28-3307 or 28-3308, except that the conviction may be used as a conviction if such conviction would be admissible had it not been set aside and may be pleaded and proved in any subsequent prosecution of such person by the state or any of its subdivisions for any offense or used by the department of transportation in enforcing the provisions of section 28-3304, 28-3306, 28-3307 or 28-3308 as if the judgment of guilt had not been set aside.
B. This section does not apply to a person convicted of a criminal offense:
1. Involving the infliction of serious physical injury.
2. Involving the use or exhibition of a deadly weapon or dangerous instrument.
3. For which the person is required or ordered by the court to register pursuant to section 13-3821.
4. For which there has been a finding of sexual motivation pursuant to section 13-118.
5. In which the victim is a minor under fifteen years of age.
6. In violation of section 28-3473, any local ordinance relating to stopping, standing or operation of a vehicle or Title 28, Chapter 3, except a violation of section 28-693 or any local ordinance relating to the same subject matter as section 28-693."
Source:
http://www.azleg.state.az.us/ars/13/00907.htm
Also of interest:
http://www.dps.state.az.us/cjsd/criminalinfosvc/courtorder.htm