The ring is mobile and thus does not have to be removed for all the
bedikot of the seven clean days, so long as you can push it to the side and check thoroughly. Ideally, one should remove the ring in order to perform one
bedikah each on days one and seven of the
shivah neki'im.
As the rules of the
hefsek taharah are more stringent, the ring should be removed for the
hefsek taharah.
Removing the ring for a short period 2-4 times during the cycle (for
hefsek taharah, two
bedikot, and mikveh) should not impair its efficacy.