Study decay around rat, say (6)
I believe the answer is:
rodent
'rat say' is the definition.
(rat is a kind of rodent)
'study decay around' is the wordplay.
'study' becomes 'den' (I've seen this before).
'decay' becomes 'rot' (rotting is a kind of decaying).
'around' is an insertion indicator.
'den' placed within 'rot' is 'RODENT'.
(Other definitions for rodent that I've seen before include "Mouse, squirrel or rat" , "Nibbler" , "Small gnawing mammal" , "Creature such as a rat" , "Vermin" .)