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Bezout orders. (English) Zbl 0826.16012

Let \(Q\) be the field of rationals or \(p\)-adic numbers, and \(K\) a subfield of the algebraic closure of \(Q\), with \((K:Q)=\infty\). Suppose that \(K\) is the quotient field of some local integrally closed domain \(R\), and \(\Lambda\) a prime PI ring with center \(R\). The authors show that \(\Lambda\) is a direct limit of classical orders over Dedekind domains. If \(R=\varinjlim R_\alpha\) with discrete valuation domains \(R_\alpha\), and \(P=\text{Rad }R\), they show that \(\Lambda\) is Bézout if and only if the \(P\)-adic completion of \(\Lambda\) is so. In the complete case, they prove that a central Bézout order \(\Lambda\) in a skewfield \(D\) consists of the elements \(d \in D\) with integral reduced norm.

MSC:

16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16K40 Infinite-dimensional and general division rings
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