Voters in North Dakota on Tuesday rejected a constitutional amendment to abolish the property tax, turning aside arguments by advocates of the measure who say the tax has proved inconsistent and is in conflict with the basic concept of property ownership.

Elsewhere Tuesday, Arizona, Virginia, Maine, Nevada, Arkansas and South Carolina held primary elections — with most of those states choosing Senate nominees — as did North Dakota, where voters also decided to let the University of North Dakota scrap its controversial nickname, the Fighting Sioux.

The property-tax-vote result, which showed North Dakotans overwhelmingly opposing the measure in unofficial returns, ended advocates' immediate hopes of making North Dakota the first in the nation to take such a step.

There, a powerful coalition of groups, including business leaders and public workers, strongly opposed the idea and raised significantly more money than the other side to spread a message that ending the property tax would mean chaos, an increase in other taxes and an end to most decision making by local city councils and county boards.

Though the property-tax ban failed, state lawmakers said they had grasped the depth of residents' frustrations and were all but certain to tackle concerns about unfair property-tax exemptions and rising assessments and tax bills.

Also in North Dakota, Rep. Rick Berg defeated businessman Duane Sand in the state's Republican primary. Berg now faces Democrat Heidi Heitkamp in the November race to replace retiring Sen. Kent Conrad. The election is expected to play a critical role in determining which party controls the Senate next year.

The vote concerning the Fighting Sioux nickname came about after the NCAA deemed it hostile and abusive, and placed the university under postseason sanctions. The state's Board of Education is now expected to retire the moniker and American Indian head logo.

Meanwhile, former Virginia Gov. and Sen. George Allen brushed aside three conservative Republican rivals in Tuesday's Virginia primary. Allen's win sets up a November clash with another former Virginia governor, Democrat Tim Kaine, in a Senate campaign closely tied to the presidential race in a state both parties consider vital for victory.

In Maine, state Sen. Cynthia Dill won the Democratic primary in the race to succeed Republican Sen. Olympia Snowe. Maine Secretary of State Charles Summers won the GOP nomination.

The front-runner, former two-term Gov. Angus King, wasn't on the ballot because he's running as an independent.

Of all the races Tuesday, the Arizona House race for the seat of former Rep. Gabrielle Giffords was the most closely watched, partly because of her compelling story of recovery from a shooting attack, and partly because holding onto the seat was important for Democrats if they want to regain control of the House.

Giffords' hand-picked Democratic candidate Ron Barber, a former Giffords aide, won the special election over Republican Jesse Kelly, who narrowly lost to Giffords in 2010 just months before she was shot. Democrats played to the senior vote by contending that Kelly would not protect Medicare and Social Security.

Barber, who was leading with 52 percent of the vote to 45 for Kelly, will complete the remainder of Giffords' term. Both candidates promised to run for a full term in the fall, setting up a possible November rematch in a redrawn district that is friendlier to Democrats.

Democrats need big gains in November to grab the majority in the House from Republicans, who now hold a 240-192 advantage with three vacancies, including Giffords' seat.

Republicans, riding high after a decisive victory in Wisconsin's gubernatorial election last Tuesday, had set their sights on Arizona, hoping a victory would give party leaders a chance to claim momentum.

Giffords, 42, resigned in January to concentrate on her recovery from a gunshot wound in January 2011 shooting rampage outside a Tucson grocery store that also injured Barber and killed six people, including a 9-year-old girl and a federal judge.