Inflexible one has no right to be promoted (4)
I believe the answer is:
iron
'inflexible' is the definition.
(I've seen this before)
'one has no right to be promoted' is the wordplay.
'one' becomes 'I' (Roman numeral).
'has' says to put letters next to each other.
'right' becomes 'R' (common abbreviation).
'to be promoted' says the letters should be written in reverse (going up, in a down clue).
'no'+'r'='nor'
'nor' in reverse letter order is 'ron'.
'i'+'ron'='IRON'
(Other definitions for iron that I've seen before include "Former dividing line across Europe" , "Box-shaped instrument of torture -- no meridian (anag)" , "stern" , "Metallic element, symbol Fe" , "inflexible > club" .)