Went on horse into the ring for the show (5)

I believe the answer is:

rodeo

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'the show' is the definition.
(rodeo is a kind of show)

I cannot understand how the remainder of the clue works.

(Other definitions for rodeo that I've seen before include "American show" , "Horse sporting event for cowboys" , "Wild West contest" , "Horse-riding competition for cowboys" , "Contest of cowboy skills" .)

I've seen this clue in The Globe and Mail.

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