On the Jacobson radical of graded rings
All commutative semigroups are described such that the Jacobson radical is homogeneous in each ring graded by .
On the Jacobson radical of strongly group graded rings
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.
A combinatorial property and power graphs of semigroups
Research on combinatorial properties of sequences in groups and semigroups originates from Bernhard Neumann's theorem answering a question of Paul Erd"{o}s. For results on related combinatorial properties of sequences in semigroups we refer to the book [3]. In 2000 the authors introduced a new combinatorial property and described all groups satisfying it. The present paper extends this result to all semigroups.
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