Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases
where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where
the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.