TOKYO — Japan’s ruling Liberal Democratic Party will select the country’s new prime minister on Friday, choosing a successor to Fumio Kishida, who reshaped Japan’s role in the international community but last month announced he would step down amid record-low approval ratings and rising prices at home.

       A record nine candidates are vying to be the leader of the LDP, and therefore the new prime minister, at a time of pronounced international volatility: The Asia-Pacific region is being reshaped by the rise of China, and the United States might reelect Donald Trump, who questioned the value of security allies like Japan during his first term as president.