On some equations related to derivations in rings.
In this paper, complex 3-dimensional Γ-graded ε-skew-symmetric and complex 3-dimensional Γ-graded ε-Lie algebras with either 1-dimensional or zero homogeneous components are classified up to isomorphism.
All commutative semigroups are described such that the Jacobson radical is homogeneous in each ring graded by .
For any non-torsion group with identity , we construct a strongly -graded ring such that the Jacobson radical is locally nilpotent, but is not locally nilpotent. This answers a question posed by Puczyłowski.