Cyclic Covers over Strongly Liftable Schemes

A smooth scheme $X$ over a field $k$ of positive characteristic is said to be strongly liftable over $W_2(k)$, if $X$ and all prime divisors on $X$ can be lifted simultaneously over $W_2(k)$. In this paper, the author gives a criterion for those cyclic covers over strongly liftable schemes that are still strongly liftable. As a corollary, cyclic covers over projective spaces of dimension at least three are strongly liftable over $W_2(k)$.     

The Tarry-Escott problem of degree two

The optimum reduced solutions of degree two of the Tarry-Escott problem over a ring of Gaussian integers and over a ring of polynomials over a finite field are completely determined.     

半环上矩阵的秩和坡上矩阵可逆的条件

研究了交换半环上矩阵的秩和坡上矩阵的可逆条件.利用Beasley的引理以及不变式,获得了交换半环上正则矩阵的行秩、列秩与Schein秩三者相等,以及坡上矩阵可逆的充要条件.推广模糊代数和分配格上矩阵的结果.     

Derivatives of probability functions and some applications

Probability functions depending upon parameters are represented as integrals over sets given by inequalities. New derivative formulas for the intergrals over a volume are considered. Derivatives are presented as sums of integrals over a volume and over a surface. Two examples are discussed: probability functions with linear constraints (random right-hand sides), and a dynamical shut-down problem with sensors.     

Homotopy properties of differential lie modules over curved coalgebras and Koszul duality

The notion of differential Lie module over a curved coalgebra is introduced. The homotopy invariance of the structure of a differential Liemodule over a curved coalgebra is proved. A relationship between the homotopy theory of differential Lie modules over curved coalgebras and the theory of Koszul duality for quadratic-scalar algebras over commutative unital rings is determined.     

Dedekind整环上的不可分解扭模

We give a direct proof to the classification of all indecomposable torsion modules over a Dedekind ring. As an application, we classify all indecomposable locally-nilpotent modules over the polynomial algebra with one variable over a field.     

On self-dual cyclic codes over finite chain rings

In this paper, we give necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite chain rings. We prove that there are no free cyclic self-dual codes over finite chain rings with odd characteristic. It is also proven that a self-dual code over a finite chain ring cannot be the lift of a binary cyclic self-dual code. The number of cyclic self-dual codes over chain rings is also investigated as an extension of the number of cyclic self-dual codes over finite fields given recently by Jia et al.     

Quasideterminant Characterization of MDS Group Codes over Abelian Groups

A group code defined over a group G is a subset of Gⁿ which forms a group under componentwise group operation. The well known matrix characterization of MDS (Maximum Distance Separable) linear codes over finite fields is generalized to MDS group codes over abelian groups, using the notion of quasideterminants defined for matrices over non-commutative rings.     

Differential equations over subalgebras of the full matrix algebra

We consider a general matrix differential equation whose parameters and solution are defined over subalgebras of the full matrix algebra. Some subalgebras are presented over which the equation splits into a set of equations over matrices of smaller orders. Multiplicative matrix equations of higher order in normal form are introduced and studied over some subalgebras of 2 × 2 matrices.     

Cleavability in Ordered Spaces

In this paper we study the role of cleavability and divisibility in the topology of generalized ordered (GO-)spaces. We characterize cleavability of a GO-space over the class of metrizable spaces, and over the spaces of irrational and rational numbers. We present a series of examples related to characterizations of cleavability over separable metric spaces and over the space of real numbers.     

Diagonalization of matrices over graded principal ideal domains

The main result of this paper is the analogue of the classical diagonal reduction of matrices over PIDs, for graded principal ideal domains. A method for diagonalizing graded matrices over a graded principal ideal domain is obtained. In Section 2 we emphasis on some applications. A procedure is given to decide whether or not a matrix defined over an ordinary Dedekind domain (i.e. nongraded), with cyclic class group, is diagonalizable. In case the answer is positive the diagonal form can be calculated. This can be done by taking a suitable graded PID which has the Dedekind domain as its part of degree zero. It turns out that, even in the case where diagonalization of a matrix over the part of degree zero is not possible, the diagonal representation over the graded ring contains useful information. The main reason for this is that the graded ring hasn't essentially more units than its part of degree zero. We illustrate this by considering the problem of von Neumann regularity of a matrix over a Gr-PID and to matrices over Dedekind domains with cyclic class group. These problems were the original motivation for studying diagonalization over graded rings.     

TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P₁,…,P_k, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements a_i over R to R whose defining polynomials are in P_i [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.     

On algebras over multicategories

We introduce a notion of a monoidal category over verbal category. In such categories we define algebras over multicategories over the same verbal categories. We also explicitly compute categories of algebras for two classes of multicategories.     

Algebras generated by two idempotents

It is shown that a noncommutative simple algebra generated over a field F by two idempotents is necessarily the ring of 2×2 matrices over a simple extension of F, and that every matrix ring over a field K can be generated over K by three idempotents.     

The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

On the coincidence of the factor and Gondran–Minoux rank functions of matrices over a semiring

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran–Minoux ranks are different.