Smart bishop not mistaken (6)

I believe the answer is:
bright
'smart' is the definition.
(synonyms)
'bishop not mistaken' is the wordplay.
I cannot really see how this works, but
'bishop' could be 'b' (abbreviation for bishop in chess) and 'b' is located in the answer.
The remaining letters 'right' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for bright that I've seen before include "7 [CLEVER]" , "bubbly?" , "Intelligent; well-lit" , "Vivid - promising - smart" , "Lustrous, shining" .)
