The upper nilradical and Jacobson radical of semigroup graded rings

Given a semigroup S  , we prove that if the upper nilradical Nil⁎(R)Nil⁎(R) is homogeneous whenever R is an S-graded ring, then the semigroup S must be cancelative and torsion-free. In case S is commutative the converse is true. Analogs of these results are established for other radicals and ideals. We also describe a large class of semigroups S with the property that whenever R is a Jacobson radical ring graded by S, then every homogeneous subring of R is also a Jacobson radical ring. These results partially answer two questions of Smoktunowicz. Examples are given delimiting the proof techniques.