Ringo, a celebrity by right (5)
I believe the answer is:
starr
'ringo' is the definition.
(Ringo Starr of the Beatles)
'a celebrity by right' is the wordplay.
'a celebrity' becomes 'star' (a star is a celebrity or famous person).
'by' means one lot of letters go next to another.
'right' becomes 'r' (common abbreviation).
'star'+'r'='STARR'
(Other definitions for starr that I've seen before include "English rock star and drummer, b. 1940" , "rush up north" , "Beatle Ringo" , "Ringo --, drummer" , "Ringo - (The Beatles)" .)