Noncommutative valuations

Schilling, O. F. G.

doi:10.1090/s0002-9904-1945-08339-6

Cited by 36 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…provided W v \θ = G is a totally ordered Abelian group (written additively) we obtain the definition of a valuation which was first proposed by Schilling in [8].…”

Section: Approximation Theorem For Dvrmentioning

confidence: 97%

“…First noncommutative valuation rings were considered and studied by O.F.G. Schilling in [8]. Below we give the main notions of a valuation and a valuation ring of a division ring.…”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

“…Definition 2.1. [8] Let G be a totally ordered group (written additively) with order relation ≥ . Add to G a special symbol ∞ such that …”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

“…In his paper [8] O.F.G. Schilling introduced the important notions of invariant valuation rings and total valuations rings and systematically studied them.…”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

See 3 more Smart Citations

Some mixed matrix problems over several discrete valuation rings

Gubareni¹

2013

J. Appl. Math. Comput. Mech.

View full text Add to dashboard Cite

Abstract. This article presents some results about several district valuation rings with a common skew field of fractions. They are obtained from the approximation theorem for discrete valuation rings. These results give the possibility to solve basic mixed matrix problems for such rings. We present the solution of some mixed flat matrix problems over several district valuation rings with common skew field of fractions.

show abstract

“…provided W v \θ = G is a totally ordered Abelian group (written additively) we obtain the definition of a valuation which was first proposed by Schilling in [8].…”

Section: Approximation Theorem For Dvrmentioning

confidence: 97%

“…First noncommutative valuation rings were considered and studied by O.F.G. Schilling in [8]. Below we give the main notions of a valuation and a valuation ring of a division ring.…”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

“…Definition 2.1. [8] Let G be a totally ordered group (written additively) with order relation ≥ . Add to G a special symbol ∞ such that …”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

“…In his paper [8] O.F.G. Schilling introduced the important notions of invariant valuation rings and total valuations rings and systematically studied them.…”

Section: Valuations and Noncommutative Dvrsmentioning

confidence: 99%

See 2 more Smart Citations

Some mixed matrix problems over several discrete valuation rings

Gubareni¹

2013

J. Appl. Math. Comput. Mech.

View full text Add to dashboard Cite

show abstract

“…In a paper entitled Noncommutative valuations, Schilling [8] 1 proved that if an algebra of finite order over its center is relatively complete in a valuation (where the value group of the nonarchimedean, exponential valuation is not assumed to be commutative) then the value group is commutative. A similar type of theorem, proved by Albert [l], states that if an algebra of finite order over a field is an ordered algebra, then the algebra is itself commutative.…”

mentioning

confidence: 99%

Nonassociative valuations

Zelinsky¹

1948

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

In a paper entitled Noncommutative valuations, Schilling [8] 1 proved that if an algebra of finite order over its center is relatively complete in a valuation (where the value group of the nonarchimedean, exponential valuation is not assumed to be commutative) then the value group is commutative. A similar type of theorem, proved by Albert [l], states that if an algebra of finite order over a field is an ordered algebra, then the algebra is itself commutative.In the present paper, the notions of valuation and ordered ring are carried over in the obvious fashion to the case of nonassociative algebras (the set of values of a valuation are no longer required to lie in an ordered group, but only in an ordered loop). In this situation, an analogue of Schilling's result remains true with an added hypothesis : If an algebra of finite order has a unity quantity and has a valuation inducing a rank one valuation of the base field, then the value loop of the algebra is commutative, associative, and archimedean-ordered. No completeness is needed. In particular, every valuation of an algebra (with a unity quantity) of finite order over an algebraic number field has a group of real numbers for its value loop.However, in general the obvious extensions of both Schilling's and Albert's results are false, since there are noncommutative, nonassociative algebras of arbitrary finite order over a field which have valuations with nonassociative value loops and which are ordered algebras. Examples of such algebras are obtained by proving that a necessary and sufficient condition for an ordered loop L to be the value loop of some algebra of finite order is that some subgroup of the center of L have finite index in L. We construct some such loops in §3. All such loops will be determined in another paper.1. Ordered loops. An ordered loop L is a set of elements x, y, z y • • • on which are defined a binary function, +, and a binary relation, >, with the following properties:(1) + is a single-valued function on LL to L.(2) If x and y are in Z, then there exist unique elements u and v in L such that u+x-y and x+v-y.

show abstract

Bemerkungen zur Theorie der bewerteten Körper

Klotzek¹,

Weinert²

1969

Mathematische Nachrichten

View full text Add to dashboard Cite

Noncommutative valuations

Cited by 36 publications

References 0 publications

Some mixed matrix problems over several discrete valuation rings

Some mixed matrix problems over several discrete valuation rings

Nonassociative valuations

Bemerkungen zur Theorie der bewerteten Körper

Contact Info

Product

Resources

About