Publish work extending on PhD with supervisor
I would suggest to mask your question by asking for advice, e.g.


  I recently wrote this paper (attach paper) and as I am still relatively new to the academic world and the basic idea for this paper originated during my PhD (see pages X,Y of my thesis), I would like to ask if I should put you as a second author.


If you already send a finished paper, they will most likely understand that you are politely asking for the ok to publish it as sole author, but you are not burning any bridges this way.