Abstract:

Rings with a completely regular generalized adjoint semigroup are characterized. It is proved that a generalized adjoint semigroup R^◇ of a ring R is a union of groups if and only if R^◇ is isomorphic to the generalized adjoint semigroup induced by E₁₁ of a Morita context 〖WTHT〗M〖WTBX〗(S,T,U,V), where S is a generalized radical ring, T  is a strongly regular ring, VU=0, and eU=Ve=0 for any idempotent e∈S.

Key words: adjoint semigroup, generalized adjoint semigroup, union of groups, generalized radical ring, idempotent element