Jacobson rings and rings strongly graded by an Abelian group (Q1110629)

Let S be a semigroup. A ring R is S-graded iff \(R=\oplus \{R_ s:\) \(s\in S\}\) where, for any \(s\in S\), \(R_ s\) is a subgroup of the additive group \((R,+)\), and \(R_ sR_ t\subseteq R_{st}\) for all \(s,t\in S\). If G is a group then R is strongly G-graded iff R is G-graded, say \(R=\oplus \{R_ g:\) \(g\in G\}\), and \(R_ gR_ h=R_{gh}\) for all \(g,h\in G\). The main result of the paper is the following Theorem 3.6. Let S be a submonoid of an abelian group G of finite torsion free rank n. Let R be a strongly G-graded ring. If \(R_ e\) (e is the identity of G) is a Jacobson ring then \(R_{[S]}=\oplus \{R_ g:\) \(g\in S\}\) is a Jacobson ring in each of the following cases: (i) \(R_ e\) is left Noetherian, (ii) \(R=R_ e[G]\) (i.e. R is the group ring of G over \(R_ e)\). That extends some results by Gilmer (on commutative semigroup rings) and Goldie and Michler (on polycyclic group rings).