Various possible versions of noncommutative valuation rings and their applications are discussed.

Noncommutative Valuation Rings

. We generalize elementary valuation theory to '-fields (division rings with involution), apply the generalized theory to the task of ordering '-fields, and give some applications to Hermitian forms.

/pdf/valuations-and-ordered-fields-2rvh8aq4qk.pdf

*-valuations and ordered *-fields

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided. Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

Rings with a distributive lattice of right ideals

Valuation Rings in Finite-Dimensional Division Algebras

/pdf/right-chain-rings-and-the-generalized-semigroup-of-30loi4pxfv.pdf

Right chain rings and the generalized semigroup of divisibility.

Bewertungen nicht kommutativer Körper

Zur Bewertungstheorie nicht kommutativer Körper

Der Approximationssatz für Bewertungen nicht kommutativer Körper

Dualität in projektiven Hjelmslevschen Räumen

Completion of valued skew fields

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