bullring vs arena

bullring and arena both are nouns.

Word	Noun	Adjective	Verb	Adverb
bullring	Yes	No	No	No
arena	Yes	No	No	No

As nouns, arena is a hypernym of bullring; that is, arena is a word with a broader meaning than bullring:

Other hypernyms of bullring include bowl, sports stadium, stadium.

bullring (noun)	arena (noun)
a stadium where bullfights take place	a playing field where sports events take place
	a large structure for open-air sports or entertainments
	the central area of an ancient Roman amphitheater where contests and spectacles were held; especially an area that was strewn with sand
	a particular environment or walk of life