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Graded rings over arithmetical orders. (English) Zbl 0536.16002
In this paper the authors study the structure of a strongly Z-graded ring R, i.e. \(R=\oplus_{n\in Z}R_ n\) with \(R_ nR_ m=R_{n+m}\) for \(n,m\in Z,\) in case \(R_ 0\) is an order in a simple Artinian ring. The following cases are considered: \(R_ 0\) is a Goldie ring, an HNP (hereditary, Noetherian, prime) ring, a maximal order, a Krull order in the sense of Chamarie, an Asano order, a Dedekind prime ring, a.s.o.. In all these cases the ring R has nice graded properties, similar to the ungraded properties of \(R_ 0\). Results of ungraded nature are also derived for R, i.e. it is shown that properties like being a Goldie ring, a maximal order, a Krull order, lift from \(R_ 0\) to R, but only go down from R to \(R_ 0\) under certain conditions. Some structural results for R are obtained in case \(R_ 0\) is an Asano order or a Dedekind prime ring. Particular attention is paid to the case when \(R_ 0\) is an HNP ring. New examples of maximal orders, Krull orders, vHC orders are obtained using generalizing Rees rings.
Reviewer: C.Năstăsescu

MSC:
16W50 Graded rings and modules (associative rings and algebras)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16Kxx Division rings and semisimple Artin rings
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