Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

The ar property and chain conditions in group rings

Abstract: It is shown that ifJ is the ring of integers or a field andG a group such that the augmentation ideal of the group ringJG has the AR property then the ringJG and the groupG satisfy certain chain conditions.

Representations of double coset hypergroups and induced representations

Abstract: The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair\(SL(2,\mathfrak{K})//SL(2,R)\) (where\(\mathfrak{K}\) are a local field andR its ring of integers) as a polynomial hypergroup on ℕ0 and characterize the (proper) subset of its dual consisting of extensible representations.

On the maximality of secret data ratio in CPTE schemes

Abstract: Based on the ring of integers modulo 2r, Chen-Pan-Tseng (2000) introduced a block-based scheme (CPT scheme) which permits in each block F of size m×n of a given binary image B to embed r =⌊log2(k+1)⌋ secret bits by changing at most two entries of F, where k=mn . As shown, the highest number of embedded secret bits for at most two bits to be changed in each block of k positions of F in any CPT-based schemes is rmax=⌊log2(1+k (k+1)/2)⌋, approximately 2r-1, twice as much as r asymptotically, and this can reached approximately in our CPTE1 scheme by using modules on the ring Z2 of integers modulo 2. A new modified scheme-CPTE2 to control the quality of the embedded blocks, in the same way as Tseng-Pan's method (2001), is established. Approximately, CPTE2 scheme gives 2r-2 embedded bits in F, twice as much as r-1 bits given by Tseng-Pan' scheme, while the quality is the same.

On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues

Abstract: Let be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let be effective nef divisors intersecting transversally in an -dimensional nonsingular projective variety . We study the degeneracy of non-Archimedean analytic maps from into under various geometric conditions. When is a rational ruled surface and and are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from into . Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over or the ring of integers of an imaginary quadratic field.

Fitting ideals of isotypic parts of even K-groups

Abstract: For a real abelian number field F with Galois group $$G=\mathrm {Gal}(F/\mathbf {Q})$$ , an odd prime p and an odd integer $$m\ge 3$$ , we study the Fitting ideal of the dual of the $$\chi $$ -part of $$\frac{K_{2m-1}(O_{F})\otimes \mathbf {Z}_{p}}{D_{p,m}(F)}$$ . Here, $$\chi $$ is a semi-simple p-adic character of G, $$K_{2m-1}(O_{F})$$ is the K-theory group of the ring of integers of F, and $$D_{p,m}(F)$$ is the submodule generated by the elements of Deligne–Soule. This Fitting ideal is then compared to the Fitting ideal of the $$\chi $$ -part of $$K_{2m-2}O_{F}$$ . Finally, an example is given, where we eliminate the dependency of the previous result on the character $$\chi $$ .