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On the Jacobson radical of graded rings. (English) Zbl 0765.16013

Let \(S\) be a commutative semigroup. Then the Jacobson radical \(J(R)\) of each semigroup-graded ring \(R=\sum_{s\in S} R_ s\) satisfies the formula \(J(R)=\sum_{s\in S}(J(R)\cap R_ s)\) if and only if \(S\) is embeddable in a torsion-free abelian group.

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16N20 Jacobson radical, quasimultiplication
20M25 Semigroup rings, multiplicative semigroups of rings
20M14 Commutative semigroups
16S35 Twisted and skew group rings, crossed products
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