Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

<Emphasis Type="Italic">q</Emphasis>-Deformations of statistical mechanical systems and motives over finite fields

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F₁ and to categorifications of Bost-Connes systems.     

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.     

关于除环的伪理想

M.K.Sen于1976年推广环的理想概念,定义了伪理想,并用依理想来研究除环、域和正则环。侯国荣1983年又推广伪理想,建立n—伪理想,得到了更好的结果。本文则证明了:非交换除环和有限域无真伪理想,因而文关于除环的几乎全部结果都是平凡的。我们还证明了非交换除环无真n—伪理想,给出了有限域无真n—伪理想的充分必要条件。文中讨论了无限域的伪理想,得到了关于除环结构的几个有用结果。     

Orderings of higher level on noetherian rings

The noncommutative algebraic geometry has found fruitful applications in quantum geometry. Similar applications are expected to be found for its younger sister the noncommutative real algebraic geometry

One of the basic results in real algebraic geometry is the Positivestellensatz. The original results of Dubois and Risler (see section 3.3 of [13]) have been extended in many directions. We refer to [14], [1], [2], [3] for commutative rings and [9], [4] for associative rings. The aim of this paper is to prove the higher level Posit ivstellensatz for noncommutative Noetherian rings. Our proof depends on the intersection theorem for orderings of higher level on skew fields ([11], Theorem 3.13). The general case of orderings of higher level on associative rings remains open.     

On schur rings of group rings of finite groups

Schur rings are rings associated to certain partitions of finite groups. They were introduced for applications in representation theory, cfr. [3][4].

The algebric structure of these rings has not been studied in depth. In this paper we determine explicit structure constants for Schur rings, we derive conditions for separability and we compute the centre. These results seem to be new even over fields.     

Waring's problem in finite rings

In this paper we obtain explicit results for Waring's problem over general finite rings, especially matrix rings over finite fields by building on analogous results over finite fields. Commutative algebra, in particular the Jacobson radical and nilpotent ideals, plays an important role in our proofs.     

Quantum cluster algebras

Cluster algebras form an axiomatically defined class of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

Algebraic properties of separated power series

We study the commutative algebra of rings of separated power series over a ring E and that of their extensions: rings of separated (and more specifically convergent) power series from a field K with a separated E-analytic structure. Both of these collections of rings already play an important role in the model theory of non-Archimedean valued fields and we establish their algebraic properties. This will make a study of the analytic geometry over such fields through the classical methods of algebraic geometry possible.     

Coordinate rings of topological Klingenberg planes II: the algebraic foundation for a projective theory

Ternary fields are the coordinate rings of affine and projective planes; however, the planes constructed over topological ternary fields are not necessarily topological. Surprisingly, the explanation of this phenomenon becomes evident in the more general theory of topological Klingenberg planes as we exhibited in [3] for the affine case. However, in the projective setting, we have a more formidable task. We must develop a new coordinate ring that admits a topological structure suitable for coordinatizing topological PK-planes. We accomplish this in two stages. In this paper, we revisit the standard coordinate rings [1, 11], discuss and resolve their deficiencies by developing a new coordinate ring as a unique extension of these refined standard rings. In a subsequent paper [4], we show that this new ring can be suitably topologized to coordinatize a topological PK-plane. This last result can then be used to explain why topological ternary fields do not necessarily coordinatize topological projective planes. Received 17 February 2000; revised 10 June 2000.     

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

Cardinal rank metric codes over Galois rings

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application.     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

On primeness of general skew inverse Laurent series ring

Ever since the introduction, skew inverse Laurent series rings have kept growing in importance, as researchers characterized their properties (such as Noetherianness, Armendarizness, McCoyness, etc.) in terms of intrinsic properties of the base ring and studied their relations with other fields of mathematics, as for example quantum mechanics theory. The goal of our paper is to study the primeness and semiprimeness of general skew inverse Laurent series rings R((x^?1;σ,δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ.     

Witt equivalence of semilocal Dedekind domains in global fields

We examine a condition for two semilocal Dedekind rings, the fields of fractions of which are global fields, to be Witt equivalent. To solve the problem we generalize the notion of a Hilbert-symbol equivalence introduced in [11] and prove that a Witt equivalence is equivalent to a Hilbert-symbol equivalence. As a result we describe a Witt equivalence in terms of field invariants.     

Self-dual codes over commutative Frobenius rings

We prove that self-dual codes exist over all finite commutative Frobenius rings, via their decomposition by the Chinese Remainder Theorem into local rings. We construct non-free self-dual codes under some conditions, using self-dual codes over finite fields, and we also construct free self-dual codes by lifting elements from the base finite field. We generalize the building-up construction for finite commutative Frobenius rings, showing that all self-dual codes with minimum weight greater than 2 can be obtained in this manner in cases where the construction applies.     

The quantum euler class and the quantum cohomology of the Grassmannians

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.