Woman not a serious socialist? (7)
I believe the answer is:
mildred
'woman' is the definition.
(woman's name)
'serious socialist?' is the wordplay.
'serious' becomes 'mild' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'socialist?' becomes 'red' (socialism is often associated with the colour red).
'mild'+'red'='MILDRED'
'not a' acts as a link.
This may not be correct. Some or all of it may be part of another bit of the clue.
(Other definitions for mildred that I've seen before include "lady?" , "Woman" , "Girl" .)