The division ring of fractions of a group ring

1983 ◽  
pp. 325-339 ◽  
Author(s):  
Robert L. Snider
Keyword(s):  
Group Ring ◽  
Division Ring ◽  
Ring Of Fractions

scholarly journals Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

2015 ◽  
Vol 25 (06) ◽  
pp. 1075-1106 ◽  
Author(s):  
Vitor O. Ferreira ◽  
Jairo Z. Gonçalves ◽  
Javier Sánchez
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Division Ring ◽  
Characteristic Zero ◽  
Free Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Ring Of Fractions ◽  
Ore Domain

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.

scholarly journals On the transcendence degree of group algebras of nilpotent groups

1984 ◽  
Vol 25 (2) ◽  
pp. 167-174 ◽  
Author(s):  
Martin Lorenz
Keyword(s):  
Nilpotent Group ◽  
Division Ring ◽  
Nilpotent Groups ◽  
Transcendence Degree ◽  
Group Algebras ◽  
Division Algebras ◽  
Free Nilpotent Group ◽  
Noetherian Domain ◽  
Ring Of Fractions ◽  
Torsion Free

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.

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.

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 Group ring based public key cryptosystems

2021 ◽  
pp. 1-22
Author(s):  
Gaurav Mittal ◽  
Sunil Kumar ◽  
Shiv Narain ◽  
Sandeep Kumar
Keyword(s):  
Group Ring ◽  
Public Key ◽  
Public Key Cryptosystems

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.

On the Lie Ring of a Division Ring

1954 ◽  
Vol 60 (3) ◽  
pp. 571 ◽  
Author(s):  
I. N. Herstein
Keyword(s):  
Division Ring ◽  
Lie Ring

scholarly journals Products of reflections in the general linear group over a division ring

1979 ◽  
Vol 28 ◽  
pp. 53-62 ◽  
Author(s):  
Dragomir Z̆. Djoković ◽  
Jerry Malzan
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
General Linear Group ◽  
General Linear
Sign in / Sign up

Export Citation Format

Share Document