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Jacobson radicals and maximal ideals of normalizing extensions applied to \(\mathbb Z\)-graded rings. (English) Zbl 0493.16002
Let \(R\) be a \(\mathbb Z\)-graded ring and let \(M\) be a graded module over \(R\). The main results of this article are: (1) the radical of \(M\) is a graded submodule of \(M\), and (2) the Brown-McCoy radical of \(R\), i.e. the intersection of all maximal (2-sided) ideals of \(R\), is a graded ideal of \(R\). The corresponding result for the Jacobson radical of \(R\) as well as the techniques used to establish (1) and (2) are due to G. Bergman [“On Jacobson radicals of graded rings”, unpublished]. The proofs proceed by viewing \(R\) as a \(\mathbb Z/n\mathbb Z\)-graded ring for different \(n\) and then using general results for finite centralizing (or normalizing) extensions to handle \(\mathbb Z/n\mathbb Z\)-graded rings. Very recently, M. Cohen and S. Montgomery have extended some of the material in this paper to \(G\)-graded rings where \(G\) is an arbitrary finite group [“Group-graded rings, smash products, and group actions”, Trans. Am. Math. Soc. 282, 237–258 (1984; Zbl 0533.16001), Addendum ibid. 300, 810–811 (1987; Zbl 0616.16001)].
Reviewer: Martin Lorenz

MSC:
16W50 Graded rings and modules (associative rings and algebras)
16Nxx Radicals and radical properties of associative rings
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References:
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