Larry Rhoden of South Dakota beat back a primary challenge on Tuesday and clinched the Republican nomination for a full term, The Associated Press said. Mr. Rhoden, a rancher, ascended to his position from lieutenant governor last year when Kristi Noem left South Dakota to join President Trump’s cabinet. Mr. Doeden finished first in the opening round of balloting last month, with several thousand more votes than Mr. Rhoden. No Democrat has held the South Dakota governorship since the 1970s. The post South Dakota Governor Seeking Full Term Clinches G.O.P.