Demure adult is first in Rome (5)

I believe the answer is:
prima
'first in rome' is the definition.
'prima' can be an answer for 'first' (I've seen this before). I am not sure about the 'in rome' bit.
'demure adult' is the wordplay.
'demure' becomes 'prim' (I've seen this in another clue).
'adult' becomes 'a' (abbreviation).
'prim'+'a'='PRIMA'
'is' is the link.
Can you help me to learn more?
(Other definitions for prima that I've seen before include "Top diva is a . . . . . donna" , "Leading" , "in pole position" , "First" .)
