Keen to rise to be prima donna (4)
I believe the answer is:
diva
'prima donna' is the definition.
('diva' can be a synonym of 'prima donna')
'keen to rise' is the wordplay.
'keen' becomes 'avid' (I've seen this in other clues).
'to rise' shows that the letters should be reversed in order.
'avid' reversed gives 'DIVA'.
'to be' acts as a link.
(Other definitions for diva that I've seen before include "Operatic soprano" , "demanding woman" , "A female opera singer" , "fat lady?" , "High-maintenace woman" .)