On \(^*\)-primes and \(^*\)-valuations. (English) Zbl 0777.16019
From the author’s introduction: Let \((D,*)\) be a *-field, it is a skew field with an anti-automorphism of order 2. For *-fields one has a notion of *-valuations. These are ordinary valuations compatible with the involution *, i.e., \(v(x^*)=v(x)\) for all non-zero \(x\in D\).
In the paper an extension theorem for abelian (abelian value group) *- valuations is given. It generalizes the extension theorem for abelian valuations on ordinary skew fields. It states: Let \(D\), \(E\) be *-fields, \(D\subset E\). An abelian *-valuation \(w\) on \(D\) extends to a *-valuation on \(E\) if and only if \(PE^ c\) is a proper ideal in \(VE^ c\), where \(V\) is the *-valuation subring of \(W\), \(P\) its maximal ideal and \(E^ c\) the commutator subgroup of \(E\).
In the paper an extension theorem for abelian (abelian value group) *- valuations is given. It generalizes the extension theorem for abelian valuations on ordinary skew fields. It states: Let \(D\), \(E\) be *-fields, \(D\subset E\). An abelian *-valuation \(w\) on \(D\) extends to a *-valuation on \(E\) if and only if \(PE^ c\) is a proper ideal in \(VE^ c\), where \(V\) is the *-valuation subring of \(W\), \(P\) its maximal ideal and \(E^ c\) the commutator subgroup of \(E\).
Reviewer: Jan Van Geel (Gent)
MSC:
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
| 16K40 | Infinite-dimensional and general division rings |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
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