TY  - JOUR
A1  - Brungs, Hans
A1  - Gräter, Joachim
T1  - Orders of Higher Rank in Semisimple Artinian Rings
Y1  - 1996
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
T1  - Algebraic elements in division rings
Y1  - 1996
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
T1  - Extending valuation rings via ultrafilters
Y1  - 1996
ER  - 
TY  - JOUR
A1  - Brungs, Hans
A1  - Gräter, Joachim
T1  - Trees and Valuation Rings
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
A1  - Klein, Markus
T1  - The Principal Axis Theorem for Holomorphic Functions
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
A1  - Weese, Martin
T1  - On the norm equation over function fields
N2  - If K is an algebraic function field of one variable over an algebraically closed field k and F is a finite extension of K, then any element a of K can be written as a norm of some b in F by Tsen's theorem. All zeros and poles of a lead to zeros and poles of b, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of b can be restricted in terms of the genus of K, respectively F. If k is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemarm surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic p > 0 using standard arguments from valuation theory
Y1  - 2004
SN  - 0024-6107
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
A1  - Wirths, Karl-Joachim
T1  - On Elementary Bounds for Sigma(infinity)(k=n)k(-s)
JF  - The American mathematical monthly : an official publication of the Mathematical Association of America
N2  - By means of elementary arguments, we derive lower and upper bounds for the infinite series Sigma(infinity)(k=n)k(-s), s is an element of R and s > 1.
Y1  - 2015
U6  - https://doi.org/10.4169/amer.math.monthly.122.02.155
SN  - 0002-9890
SN  - 1930-0972
VL  - 122
IS  - 2
SP  - 155
EP  - 158
PB  - Mathematical Assoc. of America
CY  - Washington
ER  - 
TY  - JOUR
A1  - Brungs, Hans H.
A1  - Gräter, Joachim
T1  - On central extensions of SL(2, F) admitting left-orderings
JF  - Journal of Algebra
N2  - For an arbitrary euclidean field F we introduce a central extension (G(F), Phi) of SL(2, F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order preserving bijections of the convex hull of Q in F. If F = R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2, R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.
KW  - Universal covering group
KW  - Central extensions of groups
KW  - Perfect groups
KW  - Ordered fields
KW  - Left-ordered groups
KW  - Order-preserving bijections
KW  - Euclidean fields
Y1  - 2017
U6  - https://doi.org/10.1016/j.jalgebra.2017.05.025
SN  - 0021-8693
SN  - 1090-266X
VL  - 486
SP  - 288
EP  - 327
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
T1  - Free division rings of fractions of crossed products of groups with Conradian left-orders
JF  - Forum mathematicum
N2  - Let 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 <=. For D we introduce the notion of freeness with respect to <= 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 <=. 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.
KW  - crossed product
KW  - group ring
KW  - ordered group
KW  - Conradian left-order
KW  - locally indicable group
KW  - division ring of fractions
KW  - Hughes-free
KW  - formal
KW  - power series
Y1  - 2020
U6  - https://doi.org/10.1515/forum-2019-0264
SN  - 0933-7741
SN  - 1435-5337
VL  - 32
IS  - 3
SP  - 739
EP  - 772
PB  - De Gruyter
CY  - Berlin
ER  -