Abstract: This note deals with the regularity of the adjoint semigroup of a ring. Acharacterization of adjoint regular rings is presented. For ring R which is a sum of two adjoint regular subrings, sufficient and necessary conditions for the regularity of the adjoint semigroup of R are given, which have extend the results given by Volkov and Tanana, and affirm their conjecture.

Key words: circle multiplication, adjoint semigroup, adjoint regular ring, sum of subrings