The Upper Nilradical and Jacobson Radical of Semigroup Graded Rings
نویسنده
چکیده
Given a semigroup S, we prove that if the upper nilradical 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.
منابع مشابه
On the Nilpotency of the Jacobson Radical of Semigroup Rings
Munn [11] proved that the Jacobson radical of a commutative semigroup ring is nil provided that the radical of the coefficient ring is nil. This was generalized, for semigroup algebras satisfying polynomial identities, by Okniński [14] (cf. [15, Chapter 21]), and for semigroup rings of commutative semigroups with Noetherian rings of coefficients, by Jespers [4]. It would be interesting to obtai...
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