Detective Inspector goes over each thought (4)
I believe the answer is:
idea
'thought' is the definition.
(idea is a kind of thought)
'detective inspector goes over each' is the wordplay.
'detective inspector' becomes 'DI'.
'goes over' says the letters should be written in reverse (I've seen 'going over' mean this).
'each' becomes 'ea'.
'di' back-to-front is 'id'.
'id'+'ea'='IDEA'
(Other definitions for idea that I've seen before include "Thought, belief" , "Notion, belief" , "Intention" , "Notion, image" , "Theory" .)