The bomber was identified as a man named Archiyev, police said, while the car owner, a man named Dobriyev, was detained. He is currently being questioned, police said.
The bombing is the latest in a series of terrorist attacks in the troubled North Caucasus in recent weeks.
Speaking with the president of North Ossetia, Taimuraz Mamsurov, Russian President Dmitry Medvedev expressed his condolences over the incident and said the culprits will be punished.
"We will do our best to track down those beasts who committed the terrorist act against ordinary people, a heinous terrorist act," Medvedev said. "We will do everything to find them and punish them in accordance to our country's law, and destroy them if they offer resistance."
Medvedev earlier called on officials to start investigations into the incident. Relatives of those killed and injured in the incident will receive 1 million rubles ($33,000) in compensation, while those injured will receive between 200,000 rubles ($6,500) and 400,000 rubles ($13,000), the government said.
The Vladikavkaz market has been the target of two earlier terrorist attacks. In 1999, an explosion killed 55 people and injured 300 others. In 2008, 12 people were killed and over 40 injured in a bus bombing.
Medvedev's envoy to the North Caucasus, Alexander Khloponin, is on his way to Vladikavkaz on the president's orders.
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