Categories of power stably free modules and theirK 0 groups

The conceptions of stably free order, stably free rank and power stably free isomorphism are given. The structure of power stably free module categories over a JBN ring (not necessarily commutative) and the K₀ groups of power stably free module categories over an IBN ring are investigated through a new route.     

An extension of Iwasawa's theorem on finitely generated modules over power series rings

This paper describes a structure theorem for finitely generated modules over power series rings O[[T]], where O is a maximal order in a semisimple Q_p-algebra of finite dimension over Q_p, extending Iwasawa's structure theorem (the case O=?p). A particular case of such power series ring is the ring Λ[Δ], where Λ is the power series ring ?_p?T? and Δ is a finite group of order prime to p. Several applications are given, including a new proof of a result of Iwasawa important for the relationship between Hecke characters and certain Galois representations for CM fields.     

On the ring of hurwitz series

This paper introduces the ring of Hurwitz series over a commu- tative ring with identity, and examines its structure and applications, especially to the study of differential algebra. In particular, we see that rings of Hurwitz series bear a resemblance to rings of formal power se- ries, and that for rings of positive characteristic, the structure of the ring of Hurwitz series closely mirrors that of the ground ring.     

On Polynomials Over Rickart Rings and Their Generalizations I

This paper continues the investigation of polynomials and formal power series over a ring with various annihilator conditions which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. Results of Armendariz on polynomial rings over a PP ring are extended to analogous annihilator conditions in nearrings of polynomials and nearrings of formal power series. These results are somewhat striking since, in contrast to the polynomial ring case, the nearring of polynomials or formal power series has substitution for its “multiplication” operation. These investigations provide an alternative viewpoint in illustrating the structure of polynomials and formal power series. Extensions of Rickart rings to formal power series rings are also discussed. The author was partially supported by the National Science Council, Taiwan under the grant number NSC 93-2115-M-143-001.     

On Algebraically Compact Λ-Modules

We study the structure and properties of algebraically compact modules over Λ = ? _p [[T]], the power series ring over the p-adic integer ring.     

Repeated-root cyclic and negacyclic codes over a finite chain ring

We show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

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.     

Generating series of classes of Hilbert schemes of points on orbifolds

The notion of power structure over the Grothendieck ring of complex quasi-projective varieties is used for describing generating series of classes of Hilbert schemes of zero-dimensional subschemes (“fat points”) on complex orbifolds.     

Decomposability for division algebras of exponent two and associated forms

This article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated Witt ring is torsion free.      

Integration over spaces of nonparametrized arcs and motivic versions of the monodromy zeta function

Notions of integration of motivic type over the space of arcs factorized by the natural C*-action and over the space of nonparametrized arcs (branches) are developed. As an application, two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on ℂ^d are given that correspond to these notions. Another key ingredient in the construction of these motivic versions of the zeta function is the use of the so-called power structure over the Grothendieck ring of varieties introduced by the authors. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 71–82.     

Higher Power Residue Codes and the Leech Lattice

We shall consider higher power residue codes over the ring Z₄. We will briefly introduce these codes over Z₄ and then we will find a new construction for the Leech lattice. A similar construction is used to construct some of the other lattices of rank 24.     

On the projective dimension of torsion-free abelian groups over the ring of its endomorphisms

Let E be the ring of endomorphisms of an abelian group G, and let the group G be given the structure of a left module over the ring E in the natural way. We solve the problem of determining the projective dimension of the module.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 117–124, January, 1970.     

The hermitian level of composition algebras

 The hermitian level of composition algebras with involution over a ring is studied. In particular, it is shown that the hermitian level of a composition algebra with standard involution over a semilocal ring, where two is invertible, is always a power of two when finite. Furthermore, any power of two can occur as the hermitian level of a composition algebra with non-standard involution. Some bounds are obtained for the hermitian level of a composition algebra with involution of the second kind. Received: 22 March 2002 / Revised version: 10 July 2002 Mathematics Subject Classification (2000): 17A75, 16W10, 11E25     

Ring structures on groups of arithmetic functions

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powersfα for α∈Bin(R) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.     

除环上矩阵的几个性质

该文研究了除环上的矩阵范畴，除环上n×n矩阵的柱心 幂零分解以及除环上矩阵的幂的性质     

模糊幂环

文《HX环》提出了幂环的概念。模糊数学的发展要求各种数学结构不但要由论域向其幂集上提升，而且还要求向模糊幂集上提升。本文提出Fuzzy幂环的概念，讨论Fuzzy幂环的性质与结构。     

六元gcd封闭集上Smith矩阵的整除性

我们给出了关于六元gcd封闭集S的充分必要条件,使得在整数矩阵环M_6(Z)中,定义在S上的e次幂GCD矩阵(S~e)整除e次幂LCM矩阵[S~e].这部分解决了Hong在2002年提出的一个公开问题.     

Zur Chevalley-Zerlegung von Derivationen

Let k be a valued field of characteristic zero. We consider analytic k-algebras, i.e. finite algebras over rings of convergent power series over k, and their k-derivations. The following theorem is proved: Let k be algebraically closed and A a normal analytic k-algebra of dimension 2. If there is a k-derivation on A not acting nilpotently, then A is homogeneous, i.e. a residue class ring of a power series ring by a (weighted) homogeneous ideal.The basic tool is the Chevalley decomposition of k-derivations of analytic algebras. We also use some general lemmata concerning extensions and restrictions of k-derivations which describe homogeneity.