scholarly journals Principal Primitive Ideals in Quadratic Orders and Pell’s Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-3
Author(s):  
Ahmad Issa ◽  
Hasan Sankari
Keyword(s):  
Primitive Ideal ◽  
Pell’S Equation ◽  
Primitive Ideals ◽  
Real Quadratic ◽  
Quadratic Order

In this paper, we introduce a method of determining whether the primitive ideal is principal in a real quadratic order, depending on the solvability of Pell’s equation.

scholarly journals Units of Real Quadratic Fields

1971 ◽  
Vol 44 ◽  
pp. 51-55 ◽  
Author(s):  
Akira Takaku
Keyword(s):  
Positive Solution ◽  
Quadratic Fields ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Real Quadratic

1. Let D be a positive square-free integer. Throughout this note we shall use the following notations;d = d(D): the discriminant of ,t0, u0: the least positive solution of Pell’s equation t2 — du2 = 4,

A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

1994 ◽  
Vol 37 (1) ◽  
pp. 90-96 ◽  
Author(s):  
R. A. Mollin ◽  
L.-C. Zhang ◽  
Paula Kemp
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Quadratic Field ◽  
Equivalence Theorem ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Primitive Ideals ◽  
Real Quadratic

AbstractIn this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

scholarly journals Units and Class Numbers of Real Quadratic Fields

1970 ◽  
Vol 37 ◽  
pp. 61-65
Author(s):  
Hideo Yokoi
Keyword(s):  
Quadratic Field ◽  
Integral Solution ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Pell’S Equation ◽  
Real Quadratic Fields ◽  
Real Quadratic ◽  
The Ideal

The aim of this paper is to prove the following main theorem: THEOREM. For the discriminant d>0 of a real quadratic field let (x,y) = (t,u) be the least positive integral solution of Pell’s equation x2 — dy2 = 4 and put and denote by hd the ideal class number.

scholarly journals Aperiodicity and primitive ideals of row-finite k-graphs

2014 ◽  
Vol 25 (03) ◽  
pp. 1450022 ◽  
Author(s):  
Sooran Kang ◽  
David Pask
Keyword(s):  
Primitive Ideal ◽  
Ideal Space ◽  
Gauge Invariant ◽  
C Algebra ◽  
Primitive Ideals ◽  
Spectral Spaces

We describe the primitive ideal space of the C*-algebra of a row-finite k-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the k-graph in question.

Necessary and Sufficient Conditions for the Central Norm to Equal 2h in the Simple Continued Fraction Expansion of √2hc for Any Odd c > 1

2005 ◽  
Vol 48 (1) ◽  
pp. 121-132 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Continued Fraction ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Continued Fraction Expansion ◽  
Necessary And Sufficient ◽  
Real Quadratic ◽  
Quadratic Order

AbstractWe look at the simple continued fraction expansion of √D for any D = 2hc where c > 1 is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be 2h. At the end of the paper, we also address the case where D = c is odd and the central norm of √D is equal to 2.

scholarly journals Primitive Rings of Infinite Matrices

1964 ◽  
Vol 14 (1) ◽  
pp. 47-53 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Jacobson Radical ◽  
Final Section ◽  
Matrix Ring ◽  
Primitive Ideal ◽  
Alternative Proof ◽  
Infinite Matrix ◽  
Infinite Matrices ◽  
Matrix Rings ◽  
Primitive Ideals ◽  
Infinite Case

E. C. Posner (5) has shown that a ring R is primitive if and only if the corresponding matrix ring Mn(R) is primitive. From this result he is able to deduce that the primitive ideals in Mn(R) are precisely those ideals of the form Mn(P), where P is a primitive ideal in R. This affords an alternative proof that the Jacobson radical of Mn(R) is Mn(J), where J is the Jacobson radical of R. But Patterson (3, 4) has shown that this last result does not hold in general for rings of infinite matrices and thus that the above result concerning primitive ideals cannot be extended to the infinite case. Nevertheless in this paper we are able to show that Posner's result on primitive rings does extend to infinite matrix rings. Patterson's result depends on showing that if the Jacobson radical J of R is not right vanishing then a certain matrix with entries from J does not lie in the Jacobson radical of the infinite matrix ring. In the final section of this paper we consider a ring R with this property and exhibit a primitive ideal in the infinite matrix ring, which does not arise, as above, from a primitive ideal in R. Finally the Jacobson radical of this ring is determined.

scholarly journals The prime spectrum and primitive ideal space of a graph C*-algebra

2014 ◽  
Vol 25 (07) ◽  
pp. 1450070
Author(s):  
Gene Abrams ◽  
Mark Tomforde
Keyword(s):  
Disjoint Union ◽  
Satisfying Condition ◽  
Primitive Ideal ◽  
Prime Ideals ◽  
Ideal Space ◽  
Prime Spectrum ◽  
C Algebra ◽  
Primitive Ideals

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

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