scholarly journals Free division rings of fractions of crossed products of groups with Conradian left-orders

2020 ◽  
Vol 32 (3) ◽  
pp. 739-772
Author(s):  
Joachim Gräter
Keyword(s):  
Vector Space ◽  
Endomorphism Ring ◽  
Division Ring ◽  
Skew Field ◽  
Division Rings ◽  
Products Of Groups ◽  
Ring Of Fractions ◽  
Left Order ◽  
The Right ◽  
Formal Power

AbstractLet D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.

On rings whose nonunits are a unit multiple of a nilpotent

2016 ◽  
Vol 16 (07) ◽  
pp. 1750140
Author(s):  
Peter Vámos
Keyword(s):  
Vector Space ◽  
Finite Length ◽  
Endomorphism Ring ◽  
Polynomial Identity ◽  
Morita Equivalence ◽  
Complete Characterization ◽  
Krull Dimension ◽  
Endomorphism Rings ◽  
Matrix Rings ◽  
Skew Field

The rings in the title were called UN rings by Călugăreanu in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. He gave two examples of simple UN rings: matrix rings over a skew field and a ring, which is the filtered union of such rings. We give new examples of simple UN rings as endomorphism rings of ‘vector space like’ modules and determine the structure of UN rings, which satisfy a polynomial identity or have Krull dimension. We also answer some questions in [G. Călugăreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182] about Morita equivalence of UN rings and show that this question is related to Köthe’s conjecture. Finally a complete characterization is given of modules over a Dedekind domain (in particular Abelian groups) and modules of finite length with a UN endomorphism ring.

scholarly journals On subgroups in division rings of type 2

2012 ◽  
Vol 49 (4) ◽  
pp. 549-557
Author(s):  
Bui Hai ◽  
Trinh Deo ◽  
Mai Bien
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Multiplicative Subgroups

Let D be a division ring with center F. We say that D is a division ring of type 2 if for every two elements x, y ∈ D, the division subring F(x, y) is a finite dimensional vector space over F. In this paper we investigate multiplicative subgroups in such a ring.

scholarly journals A note on multiplicative commutators of division rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950031
Author(s):  
Roozbeh Hazrat
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Commutator Subgroup ◽  
Division Rings

We give an example of a division ring [Formula: see text] whose multiplicative commutator subgroup does not generate [Formula: see text] as a vector space over its center, thus disproving the conjecture posed in [M. Aghabali, S. Akbari, M. Ariannejad and A. Madadi, Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12(8) (2013) 7 pp.].

scholarly journals LINEAR DIVISION RINGS

2009 ◽  
Vol 12 (17) ◽  
pp. 5-11
Author(s):  
Bien Hoang Mai ◽  
Hai Xuan Bui
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Finite Subset ◽  
Multiplicative Group ◽  
Dimensional Vector ◽  
Finitely Generated ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

Decompositions of Linear Transformations over Division Rings

2012 ◽  
Vol 19 (03) ◽  
pp. 459-464 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Linear Transformations ◽  
Division Rings

Let V be a countably generated right vector space over a division ring D. If |D| ≠ 2,3, then for any γ ∈ End D(V), there exist α, β ∈ Aut D(V) such that γ-α, γ-α-1, γ2-β2 ∈ Aut D(V).

Groups definable in ordered vector spaces over ordered division rings

2007 ◽  
Vol 72 (4) ◽  
pp. 1108-1140 ◽  
Author(s):  
Pantelis E. Eleftheriou ◽  
Sergei Starchenko
Keyword(s):  
Fundamental Group ◽  
Vector Space ◽  
Division Ring ◽  
Quotient Group ◽  
Vector Spaces ◽  
Division Rings ◽  
Ordered Vector Space ◽  
Dimensional Group ◽  
Ordered Vector Spaces

AbstractLet M = 〈M, +, <, 0, {λ}λЄD〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a ‘definable quotient group’ U/L, for some convex V-definable subgroup U of 〈Mn, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L.

scholarly journals The universality of Hughes-free division rings

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrei Jaikin-Zapirain
Keyword(s):  
Division Ring ◽  
Amenable Group ◽  
Crossed Product ◽  
Division Rings ◽  
Ring Of Fractions ◽  
Open Question ◽  
Indicable Group

AbstractLet $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.

scholarly journals Approximating pointwise products of quasimodes

2020 ◽  
Vol 32 (3) ◽  
pp. 541-552
Author(s):  
Mei Ling Jin
Keyword(s):  
Vector Space ◽  
Harmonic Analysis ◽  
Riemannian Manifolds ◽  
Spectral Projection ◽  
Beltrami Operator ◽  
Singular Potentials ◽  
Link Type ◽  
Right Hand ◽  
Low Degree ◽  
The Right

AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

scholarly journals SOURCES OF THE POWER AND THE AUTHORITY OF MANAGERS

2018 ◽  
Vol 28 (1) ◽  
pp. 79-84
Author(s):  
Vojo Belovski ◽  
Biljana Todorova
Keyword(s):  
Interpersonal Relationships ◽  
Religious Media ◽  
The Status ◽  
The Will ◽  
The Republic ◽  
Intellectual Knowledge ◽  
The Right ◽  
And Behavior ◽  
Formal Power ◽  
Power And Authority

The paper starts from the general approach to the content and essence of the categories of power and authority and their interrelationship at the level of theoretical analysis and practical existence and manifestation.The sources from which the power and the authority of managers emerge will be analyzed taking into account their position and role in the organizations and other forms of the existence of the managerial function.The power is the right to order and obligation to respect / apply the order - it is very present in the work and behavior of the managers. The power is visible in the area of the state activities, in the education system, among the family.The authority represents carrying out the will even when it is contrary to the interests of others. You can talk about economic, ideological, religious, media authority, the authority of political parties and interest groups.Organizations are composed of persons who perform greater or lesser degrees of authority and power. Sometimes the power and authority in the organization arise from the position of a person in the organization or from the knowledge and skills that a person possesses. Others express their authority in interpersonal relationships through their character. In practice, it is seen that individuals have formal power and no real authority.Most directly, the authority of managers is derived from their functions / activities in the enterprise, from the right to command and direct other people in their tasks and responsibilities. Their power stems from the right and the ability to create an environment in which other individuals will participate in the realization of the organization's goals, in other words, the right to create an atmosphere that will encourage people to dedicate themselves to the work and development of the enterprise.The authority of managers arises from their intellectual knowledge, often higher than the knowledge of employees, which also activates authority as a voluntary acknowledgment of influence on the subordinate.Through an analytical approach, analyzes will be made on some issues and aspects of the status of managers in the Macedonian society, through projected grouping / classification of types of managers. Also, an answer to the question of why the managerial function in the Republic of Macedonia is reviving.

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.

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