On horseback, Ruth is arrogant (6)
I believe the answer is:
uppity
'arrogant' is the definition.
(I've seen this in another clue)
'on horseback ruth' is the wordplay.
'on horseback' becomes 'up' (up can mean up on a horse).
'ruth' becomes 'pity' (synonyms).
'up'+'pity'='UPPITY'
'is' acts as a link.
(Other definitions for uppity that I've seen before include "Impertinent" , "Cheeky or presumptuous" , "Informally pert or arrogant" , "Bumptious" , "Needing to be slapped down?" .)