scholarly journals Normalité de certains anneaux déterminantiels quantiques

1999 ◽  
Vol 42 (3) ◽  
pp. 621-640 ◽  
Author(s):  
Laurent Rigal
Keyword(s):  
Residue Class ◽  
Quantum Algebras ◽  
Regular Functions ◽  
Maximal Orders ◽  
Ring Of Fractions ◽  
Prime Factors ◽  
The Matrix ◽  
Residue Class Ring ◽  
Class Ring ◽  
The Ideal

Let Kq[X] = Oq(M(m, n)) be the quantization of the ring of regular functions on m × n matrices and Iq (X) be the ideal generated by the 2 × 2 quantum minors of the matrix X=(Xij)l≤i≤m, I≤j≤n of generators of Kq[X]. The residue class ring Rq(X) = Kq[X]/Iq(X) (a quantum analogue of determinantal rings) is shown to be an integral domain and a maximal order in its divisionring of fractions. For the proof we use a general lemma concerning maximalorders that we first establish. This lemma actually applies widely to prime factors of quantum algebras. We also prove that, if the parameter isnot a root of unity, all the prime factors of the uniparameter quantum space are maximal orders in their division ring of fractions.

scholarly journals Sums of generators of ideals in residue class ring

2017 ◽  
Vol 174 ◽  
pp. 14-25 ◽  
Author(s):  
Xin Zhang ◽  
Chun-Gang Ji
Keyword(s):  
Residue Class ◽  
Residue Class Ring ◽  
Class Ring

Simple reduction theorems for extension and torsion functors

1961 ◽  
Vol 57 (3) ◽  
pp. 483-488
Author(s):  
D. G. Northcott
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Residue Class ◽  
Identity Element ◽  
Negative Integer ◽  
Reduction Theorem ◽  
Residue Class Ring ◽  
Image Position ◽  
Class Ring ◽  
Simple Reduction

Let Λ be a commutative ring with an identity element and c an element of Λ which is not a zero divisor Denote by Ω the residue class ring Λ/Λc. If now M is a Λ-module for which c is not a zero divisor, and A is an Ω-module, then a theorem of Rees (2) asserts that, for every non-negative integer n, we have a Λ-isomorphismThis reduction theorem has found a number of useful and interesting applications.

scholarly journals New Generalized Cyclotomic Quaternary Sequences with Large Linear Complexity and a Product of Two Primes Period

Information ◽  
2021 ◽  
Vol 12 (5) ◽  
pp. 193
Author(s):  
Jiang Ma ◽  
Wei Zhao ◽  
Yanguo Jia ◽  
Haiyang Jiang
Keyword(s):  
Spectral Sequence ◽  
Linear Complexity ◽  
Residue Class ◽  
Important Criterion ◽  
Random Sequences ◽  
Gray Mapping ◽  
Quaternary Sequences ◽  
Hamming Weights ◽  
Residue Class Ring ◽  
Class Ring

Linear complexity is an important criterion to characterize the unpredictability of pseudo-random sequences, and large linear complexity corresponds to high cryptographic strength. Pseudo-random Sequences with a large linear complexity property are of importance in many domains. In this paper, based on the theory of inverse Gray mapping, two classes of new generalized cyclotomic quaternary sequences with period pq are constructed, where pq is a product of two large distinct primes. In addition, we give the linear complexity over the residue class ring Z4 via the Hamming weights of their Fourier spectral sequence. The results show that these two kinds of sequences have large linear complexity.

scholarly journals Permutation polynomials in several variables over residue class rings

1987 ◽  
Vol 43 (2) ◽  
pp. 171-175 ◽  
Author(s):  
H. K. Kaiser ◽  
W. Nöbauer
Keyword(s):  
Commutative Ring ◽  
Residue Class ◽  
Polynomial Function ◽  
Permutation Polynomial ◽  
Permutation Polynomials ◽  
Several Variables ◽  
Residue Class Ring ◽  
Class Ring

AbstractThe concept of a permutation polynomial function over a commutative ring with 1 can be generalized to multiplace functions in two different ways, yielding the notion of a k-ary permutation polynomial function (k > 1, k ∈ N) and the notion of a strict k-ary permutation polynomial function respectively. It is shown that in the case of a residue class ring Zm of the integers these two notions coincide if and only if m is squarefree.

scholarly journals A characterisation of PSL2(Zþλ) and PGL2(Zþλ)

1968 ◽  
Vol 8 (3) ◽  
pp. 523-543 ◽  
Author(s):  
G. E. Wall
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Residue Class ◽  
Linear Groups ◽  
Cyclic Subgroups ◽  
Even Order ◽  
Residue Class Ring ◽  
Projective Linear Groups ◽  
Class Ring

Let Fq denote the finite field with q elements, Zm the residue class ring Z/mZ. It is known that the projective linear groups G = PSL2(Fq) and PGL2(Fq) (q a prime-power ≥ 4) are characterised among finite insoluble groups by the property that, if two cyclic subgroups of G of even order intersect non-trivially, they generate a cyclic subgroup (cf. Brauer, Suzuki, Wall [2], Gorenstein, Walter [3]). In this paper, we give a similar characterisation of the groups G = PSL2 (Zþt+1) and PGL2 (Zþt+1) (p a prime ≥ 5, t ≥ 1).

scholarly journals GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

2004 ◽  
Vol 47 (1) ◽  
pp. 163-190 ◽  
Author(s):  
Stéphane Launois
Keyword(s):  
Prime Ideal ◽  
Subject Classification ◽  
Prime Ideals ◽  
Generating Set ◽  
Maximal Orders ◽  
The Matrix ◽  
Secondary 16W35 ◽  
The Ideal

AbstractIt is known that, for generic $q$, the $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, J. Alg., in press). In this paper, $m$ and $p$ being given, we construct an algorithm which computes a generating set of quantum minors for each $\mathcal{H}$-invariant prime ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. We also describe, in the general case, an explicit generating set of quantum minors for some particular $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. In particular, if $(Y_{i,\alpha})_{(i,\alpha)\in[[1,m]]\times[[1,p]]}$ denotes the matrix of the canonical generators of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, we prove that, if $u\geq3$, the ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ generated by $Y_{1,p}$ and the $u\times u$ quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc.46 (2003), 513–529).AMS 2000 Mathematics subject classification: Primary 16P40. Secondary 16W35; 20G42

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