t次正规环的一个恒等式定理及其应用

一个适合 t 正规多项式的环叫做 t 次正规环。本文得到一个关于 t 次正规环的恒等式定理,并在此定理基础上得出交换环的另一些恒等式,它们是著名的环的交换性问题的一些结果或类似的结果。     

Amitsur定理和Regev定理的推广

本文将PI一环论中关于恒等式和中心多项式的Amitsur定理和Regev定理同时由域推广到无零因子环,得到无零因子环上全矩阵环的两个相应定理。     

素GPI-环的幂零多项式

R是素 GPI-环 ,若多项式 f (x1,… ,xd)在 R上是幂零的 ,则或 f (x1,… ,xd)是 R的恒等式 ,或 R是有限域上的有限矩阵环 .     

交换环上全矩阵代数迹恒等式的几个定理

本文研究了交换环上全矩阵代数的迹恒等式，特别研究了积的迹为零的多项式，它好比矩阵或多项式正交，在现代物理学中有着十分广泛的应用．     

环上带有广义导子和自同构的恒等式

设R是一个素环,RF是它的左Martindale商环.如果φ(xigik△ij)是环R的某个本质理想I的一个多重线性既约且带有自同构的广义微分恒等式,那么φ(zikj)是环RF的一个广义多项式恒等式.设R是一个具有特征P≥0的半素环,RF是它的左Martindale商环.如果φ(xigik△ijfik)是环R的一个多重线性既约且带有自同构的广义微分恒等式,那么φ(zikjfike(△ij)是环RF的一个广义多项式恒等式,这里fik和e(△ij)是RF中的幂等元.     

交换环上全矩阵代数的迹恒等式

本文研究了交换环上全矩阵代数的迹恒等式，特别研究了积的迹为零的多项式．     

环R_(k,m)上的线性码及其MacWilliams恒等式

研究了环R_(k,m)=Fq[u,v]/〈uk,vm,uv-vu〉上的线性码及其MacWilliams恒等式,其中q是素数p的方幂且k≥m≥1.首先给出了R_(k,m)到Fkmq的Gray映射,此映射关于Lee重量具有保距性和保对偶性,然后证明了环R_(k,m)上线性码相应重量计数多项式的MacWilliams恒等式,特别地给出了环R_(k,m)上线性码关于Lee重量计数多项式的MacWilliams恒等式.     

多项式剩余类环上循环码新的表示

在编码理论中，多项式剩余类环是非常有意义的，它已经用来构造最优频率希望序列。本文，定义了多项式剩余类环上循环码的离散傅立叶变换及Mattson-Solomon（MS）多项式，证明了多项式剩余类环上的循环码同构于多项式剩余类环的Galois扩张的理想。     

McCoy环的Ore扩张(英文)

本文研究了斜多项式环与微分多项式环的McCoy性质,证明了如果环R是α-compatible和可逆的,那么斜多项式R[x;α]是McCoy环当且仅当环R是McCoy环;同时我们也证明了如果环R是δ-compatible与可逆的,那么微分多项式环R[x;δ]是McCoy环当且仅当环R是McCoy环.因此本文对McCoy环的相关结论进行了推广.     

正规几乎PP环

首先探讨正规几乎PP环的内刻划及左右对称性；其次，研究正规几乎PP环与PP环的关系，最后证明多项式环R［X」是正规几乎PP环当且仅当R是正规几乎PP环．     

两个包含欧拉数的恒等式的等价性(英文)

基于n维多项式空间中基之间的线性变换,证明了两个包含欧拉数的恒等式是等价.     

A Generalization of a theorem of chang

Szigeti, Tuza and Révész have developed a method in [6] to obtain polynomial identities for the n×n matrix ring over a commutative ring starting from directed Eulerian graphs. These polynomials are called Euler-ian. In the first part of this paper we show some polynomials that are in the T-ideal generated by a certain set of Eulerian polynomials, hence we get some identities of the n×n matrices. This result is a generalization of a theorem of Chang [l]. After that, using this theorem, we show that any Eulerian identity arising from a graph which lias d-fold multiple edges follows from the standard identity of degree d     

Euler–Mahonian statistics via polyhedral geometry

A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler–Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler–Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.     

Linear identities and commuting maps in rings with involution

Abstract. In this paper we extend Martindale's result by showing that the symmetric elements or the skew elements of a prime ring with involution of characteristic not 2 do not satisfy certain linear generalized polynomial identities. As applications, we determine the centralizing additive maps and the commuting traces of biadditive maps on the symmetric elements of a prime ring.     

Skew Schubert functions and the Pieri formula for flag manifolds

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.

     

Combinatorial methods in the theory of Cohen-Macaulay rings

Characterizations of Cohen-Macaulay posets are given in terms of the nonsingularity of certain incidence matrices. These results are applied to derive a purely combinatorial construction of basic systems for Stanley-Reisner rings of shellable posets. A procedure for transferring identities from one polynomial ring to another is then used to obtain basic systems for partition rings.     

多项式矩阵代数性质讨论

本文利用系统与控制论中有关多项式矩阵的结果,对多项式矩阵代数性质进行讨论,得到的主要结果有多项式方阵环是主理想环,也是主单侧理想环。     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

q-Pell sequences and two identities of V.A. Lebesgue

We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.     

Riordan阵与Cauchy多项式

给出了Cauchy多项式c_n~α(z)的定义,并导出它的生成函数.再利用Riordan阵方法得到包含Cauchy多项式的一些恒等式,获得它与广义调和多项式H_n~((r))(z),广义Stirling多项式P_(n,r)(z)的关系式.