Singular elements and the Witt equivalence of rings of algebraic integers

2008 ◽  
Vol 17 (2) ◽  
pp. 185-217
Author(s):  
Beata Rothkegel ◽  
Alfred Czogała
Keyword(s):  
Algebraic Integers ◽  
Witt Equivalence ◽  
Rings Of Algebraic Integers

scholarly journals Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

2020 ◽  
Author(s):  
Chris Bruce
Keyword(s):  
Phase Transitions ◽  
Class Group ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
C Algebras ◽  
Rings Of Algebraic Integers ◽  
Beta 2 ◽  
Factor State ◽  
Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

scholarly journals Matrices over rings of algebraic integers

1991 ◽  
Vol 145 ◽  
pp. 1-20 ◽  
Author(s):  
Morris Newman ◽  
Robert C. Thompson
Keyword(s):  
Algebraic Integers ◽  
Rings Of Algebraic Integers

K i Groups of Rings of Algebraic Integers

1975 ◽  
Vol 101 (1) ◽  
pp. 20 ◽  
Author(s):  
B. Harris ◽  
G. Segal
Keyword(s):  
Algebraic Integers ◽  
Rings Of Algebraic Integers
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