Actually Mongolia was effectively annexed by BOTH China AND Russia. After the fall of the Mongol Empire, the Mongols mostly reverted back to their nomadic life styles. However, the area fell under Chinese rule from 1636 through 1691. The Chinese weren't exactly direct rulers of the Mongols, the Mongolians were a kind of vassal state to the Chinese, similar to Korea at the time, but the Chinese at least nominally ruled Mongolia until collapse of the Empire in 1911.

After the last Chinese Emperor was desposed, the Mongols leader, Bogd Khaan, declared independence, which put the Mongols into conflict with the new Republic of China. This conflict continued until the Russian revolution literally spilled over the border. Mongolia defeated the Chinese forces in 1921 with the help of the USSR's communist forces.

Officially, Mongolia was it's own seperate and independent nation. Unofficially, things were lot more complicated. The USSR heavily pressured Mongolia into adopting a communist government, the People's Republic of Mongolia, which remained in a similar position as the Eastern Block nations of the Iron Curtain until the collapse of Communism in 1989.