If you keep cutting it down.

i hear this is an aggressive tree species so i want to know if roots can survive and flourish on their own after the tree is gone?

Rhus - sumac species - are colonizers. If a well established plant, it will be sending up suckers from the root system and any of those that produce above ground growth will continue to feed the root system. Unless you remove ALL of them promptly, yes, the roots can and will flourish on their own.

thank you very much for the intuitive explanation.

Very good info. I have attempted to find all or most of the root structures and cut them. Will the remaining roots still sprout suckers even in they are detached from the tree. Many thanks as I am in a mess!!!! Nova Scotia, Canada