scholarly journals A class of hyperrings and hyperfields

1983 ◽  
Vol 6 (2) ◽  
pp. 307-311 ◽  
Author(s):  
Marc Krasner
Keyword(s):  
Normal Subgroup ◽  
Multiplicative Group ◽  
Multiplicative Semigroup ◽  
Technical Tool ◽  
Valued Fields ◽  
Commutative Theory

Hyperring is a structure generalizing that of a ring, but where the addition is not a composition, but a hypercomposition, i.e., the sumx+yof two elements,x,y, of a hyperringHis, in general, not an element but a subset ofH. When the non-zero elements of a hyperring form a multiplicative group, the hyperring is called a hyperfield, and this structure generalizes that of a field. A certain class of hyperfields (residual hyperfields of valued fields) has been used by the author [1] as an important technical tool in his theory of approximation of complete valued fields by sequences of such fields. Tne non-commutative theory of hyperrings (particularly Artinian) has been studied in depth by Stratigopoulos [2].The question arises: How common are hyperrings? We prove in this paper that a conveniently defined quotientR/Gof any ringRby any normal subgroupGof its multiplicative semigroup is always a hyperring which is a hyperfield whenRis a field. We ask: Are all hyperrings isomorphic to some subhyperring of a hyperring belonging to the class just described?

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

Canonical Submersion and Lie Homomorphisms Normal Subgroup

2020 ◽  
Vol 23 (1) ◽  
pp. 97-101
Author(s):  
Mikhail Petrichenko ◽  
Dmitry W. Serow
Keyword(s):  
Normal Subgroup ◽  
Canonical System ◽  
System Of Equations ◽  
Core Group ◽  
The Core ◽  
Switch Group

Normal subgroup module f (module over the ring F = [ f ] 1; 2-diffeomorphisms) coincides with the kernel Ker Lf derivations along the field. The core consists of the trivial homomorphism (integrals of the system v = x = f (t; x )) and bundles with zero switch group Lf , obtained from the condition ᐁ( ω × f ) = 0. There is the analog of the Liouville for trivial immersion. In this case, the core group Lf derivations along the field replenished elements V ( z ), such that ᐁz = ω × f. Hence, the core group Lf updated elements helicoid (spiral) bundles, in particular, such that f = ᐁU. System as an example Crocco shown that the canonical system does not permit the trivial embedding: the canonical system of equations are the closure of the class of systems that permit a submersion.

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

A Study On Technical Analysis With Reference To International Forex

Think India ◽  
2019 ◽  
Vol 22 (3) ◽  
pp. 1129-1144
Author(s):  
Bichith C. Sekhar ◽  
A. Umamaheswari
Keyword(s):  
Foreign Exchange ◽  
Abnormal Return ◽  
Foreign Exchange Market ◽  
Relative Strength ◽  
Exchange Market ◽  
Strength Index ◽  
Technical Tool ◽  
Alpha Beta ◽  
Decentralized Market ◽  
Currency Conversion

The foreign exchange market (Forex, FX, or currency market) is a global decentralized market for the trading of currencies. The foreign exchange market assists international trade and investments by enabling currency conversion. Our study is to test the technical tools to analyze about the technical impact and its return in the market.  For this purpose 13 cross currency pairs were taken as sample size and Jensen’s Alpha, Beta, Relative Strength Index, and Buy and Hold Abnormal Return were used as technical tool for analysis and the conclusion is that it’s not preferred to invest in JPY pairs as the volatility and the return are not up to the mark and its preferred to invest in EURCAD as the return was high when compared to other scripts and the market was moving accordingly to its cross currency pair.

Zwischenevaluation des Wald-Wild-Management-Instruments (WWMI) im Pilotprogramm effor2 «Wald und Wild» | Preliminary evaluation of the Game-Forest-Management-Tool (GFMT) in the effor2 pilot programme «Game and Forest»

2003 ◽  
Vol 154 (8) ◽  
pp. 305-313 ◽  
Author(s):  
Roman Eyholzer ◽  
Martin Baumann ◽  
Rolf Manser
Keyword(s):  
Forest Management ◽  
Management Practice ◽  
Management Tool ◽  
Preliminary Evaluation ◽  
Optimal Contract ◽  
Technical Tool ◽  
Entire Study ◽  
Pilot Programme ◽  
Complex Problems ◽  
Entire Study Area

Faced with the challenging task of balancing forest interests and wildlife, the Swiss Forest Agency initiated the pilot programme«Game and Forest», which is committed to a philosophy of goal-oriented management practice and a redesign of forestry subsidizing. Within this programme the diverse goals of forestry and hunting have been amalgamated to a superimposed goal and set out in a corresponding contract. The Game-Forest-Management-Tool (GFMT) has been divised to simulate the effect of various strategies to deal with the complex problems of forest-wildlife. Optimal contract-fulfilling procedures can be simulated on a PC using this technical tool. The efficiency of the measures suggested by simulations that were carried out are being tested in a study area within the pilot programme, «Game and Forest». Half way through this trial, after two years, we can say that there has been no significant increase of non-browsed areas. In 2004, after the collection of data for the entire study area, we will be able to tell whether applying this computer simulated strategy truly leads to an augmentation of non-browsed area and a decrease in bark-peeled forests in the pilot area.

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