Abstract: This paper deals with the relationship between a ring and its circle semigroup. Adjoint clifford rings and the rings with small circle semigroups are characterized in terms of circle multiplication, and a commutative adjoint Clifford ring is constructed in which Jacobson radical is not a summand so as to confirm a conjecture of Heatherly and Tucci.

Key words: circle semigroup, adjoint Clifford ring, small monoid