关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

确定有限域上给定周期的不可约多项式的个数以及利用低次不可约多项式构造高次不可约多项式

主要利用较献[4]更为简明的方法证明了有关有限域Fq(q为一个素数幂）上的以l为周期的n次不可约多项式的个数的结论。另外，本结合结合初等数论知识得到了前面这个结论的几个推论，并对利用低次不可约多项式构造高次不可约多项式进行了研究。     

关于多项式因式分解的两个定理

关于多项式因式分解的两个定理彭明海（湖南吉首大学４１６０００）我在《高等代数》教学中，发现下面两个定理，今介绍出来，供同行参考．定理１设ｆ（ｘ）＝ａｎｘｎ＋ａｎ－ａｎ－１ｎ＋…＋（Ｚ［ｘ］表示整系数多项式集合）如果有一个素数ｐ满足条件且则ｆ（ｘ）在有...     

关于实系数代数多项式根的分布

给定一实系数多项式:p(z)=a_0z~n a_1z~(n-1) … a_n,不失一般性,假定a_0>0.本文主要给出有关多项式(1)的根的分布的结果.定理1 如果系数{a_i}_i=0~m满足条件△~2a_k≥0,k=0,1,2,…,(n-1),其中△~2 a_k是二级差分,那么多项式(1)的所有根位于圆|z|<1外.     

二元整系数多项式因式分解的一种理论和算法

我们发现可以把二元多项式盾成系数为一元多项式的一元多项式来进行分解，据此，本文建立了二元整系数多项式因式分解的一种理论，提出了一个完整的分解二元整系数多项式的算法。这个算法还能很自然地推广成分解多元整系数多项式的算法。     

多元整系数多项式因式分解的一种理论和算法

本文将ｔ（ｔ是大于２的整数）元整系数多项式看成为系数为ｔ－２元整系数多项式的二元多项式，建立了多元整系数多项式因式分解的一种新理论，进而得到了分解多元整系数多项式的一个有力的算法。     

对称多项式概念的推广

定义一类G-对称多项式。它是对于Sn的子群G中置换不变的多项式,当G为Sn或一个n轮换生成的循环子群时,相应的G-对称多项式就是对称多项式或轮换对称多项式。     

推广的Kantorovich多项式的一些基本性质

本文对推广的Bernstein-Kantorovich多项式进行了深入地研究和讨论,给出并证明了一些重要的基本性质.     

关于几类图族伴随多项式的第四项系数

主要研究了几类图族伴随多项式第四项系数的规律，此结果有助于进一步讨论这些图族补图的色唯一性、色等价划分．     

关于多项式函数的计数

Polynomial functions （in particular, permutation polynomials） play an important role in the design of modern cryptosystem. In this note the problem of counting the number of polynomial functions over finite commutative rings is discussed. Let A be a general finite commutative local ring. Under a certain condition, the counting formula of the number of polynomial functions over A is obtained. Before this paper, some results over special finite commutative rings were obtained by many authors.     

局部环上幂等矩阵线性组合广义逆之间的关系

设R是2为单位的局部环.研究了R上三个两两可换的n阶非零幂等矩阵的线性组合广义逆之间的包含关系,确定了R上一类特殊矩阵广义逆的列表算法.利用这种列表算法和相关的矩阵理论,得到了这些矩阵线性组合广义逆之间的包含关系的充要条件,推广了矩阵自反广义逆的逆反律的相关结果.     

Fuzzy理想和Fuzzy幂零根

本文给出了Ｆ—准素（ＦＷ—准素）理想的有限交是Ｆ－准素理想（ＦＷ－准素理想）的几个充分条件。此外，证明了［５］中定义的一个交换环Ｒ的Ｆｕｚｚｙ幂零根是Ｒ的幂零根Ｋ的特征函数Ｘ＿Ｋ。     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

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.     

When Are Weak Permutation Polynomials Strong?

For a commutative finite ring with identity R, the two definitions of "permutaIion polynomial in several indeterminates over R" coincide if and only if R is a direct sum of finite fields.     

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.     

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.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

环Z/p~kZ上矩阵广义逆的拓展

设R=Z/pkZ(其中k>1,p是一个奇素数),A是R上一个给定的可相似对角化的n阶矩阵.利用组合方法和有限局部环上的矩阵方法,讨论了矩阵A的拓展广义逆,得到了矩阵A的拓展广义逆存在的充要条件和一些的计数定理.