On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

On Prime Rings Satisfying a Certain Generalized Polynomial Identity

Abstract

Recently, Beidar, Fong and Bokut proved that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field. We shall extend their result to the case when generalized polynomial contains three summands.     

On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity

Let R be a ring satisfying a polynomial identity, and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D]) ∩ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative.     

On Primitive Ideals in Polynomial Rings over Nil Rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R. Presented by S. MontgomeryMathematics Subject Classifications (2000) 16N40, 16N20, 16N60, 16D25.Agata Smoktunowicz: Current address: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw 10, niadeckich 8, P.O. Box 21, Poland.     

Derivations and Skew Polynomial Rings

Soient D un corps non nécessairement commutatif et L un sous-corps de D. On établit une condition nécessaire et suffisante pour que le groupe multiplicatif L de L soit d'indice fini dans son normalisateur N dans D. Lorsque la dimension à gauche [D : L]_g est un nombre premier, on précise le groupe N/L et la structure de D.     

Polynomial Rings over Symmetric Rings Need Not Be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

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。     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

Poisson Prime Ideals of Poisson Polynomial Rings

Let A be a finitely generated Poisson algebra over a field of characteristic zero. Here we prove that every Poisson prime ideal of A is prime and give a method to find all Poisson prime ideals in an arbitrary Poisson polynomial ring A[x; α, δ].     

Closed Polynomials in Polynomial Rings over Unique Factorization Domains

Let R be a UFD, and let M(R, n) be the set of all subalgebras of the form R[f], where f ∈ R[x ₁,…, x _n ]?R. For a polynomial f ∈ R[x ₁,…, x _n ]?R, we prove that R[f] is a maximal element of M(R, n) if and only if it is integrally closed in R[x ₁,…, x _n ] and Q(R)[f] ∩ R[x ₁,…, x _n ] = R[f]. Moreover, we prove that, in the case where the characteristic of R equals zero, R[f] is a maximal element of M(R, n) if and only if there exists an R-derivation on R[x ₁,…, x _n ] whose kernel equals R[f].     

Integer-Valued Polynomials over Matrix Rings

When D is a commutative integral domain with field of fractions K, the ring Int(D) = {f ∈ K[x] | f(D) ? D} of integer-valued polynomials over D is well-understood. This article considers the construction of integer-valued polynomials over matrix rings with entries in an integral domain. Given an integral domain D with field of fractions K, we define Int(M _n (D)): = {f ∈ M _n (K)[x] | f(M _n (D)) ? M _n (D)}. We prove that Int(M _n (D)) is a ring and investigate its structure and ideals. We also derive a generating set for Int(M _n (?)) and prove that Int(M _n (?)) is non-Noetherian.     

On *-Clean Group Rings II

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C₃, C₄, S₃, and Q₈. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebra FC_p is *-clean, where F is a field and C_p is a cyclic group of prime order p. In this article, we extend the above mentioned result from FC_p to F_qC_p^k, where F_q is a finite field and C_p^k is a cyclic group of an odd prime power order p^k. For the general case when G = C_n is cyclic group of order n, we also provide a necessary condition and a few sufficient conditions for F_qC_n to be *-clean.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

关于斜McCoy环

For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

Polynomial Rings Over Symmetric Rings Need Not be Symmetric

Let R be a ring with identity. The polynomial ring over R is denoted by R[x] with x its indeterminate. It is shown that polynomial rings over symmetric rings need not be symmetric by an example.     

On Rings Having McCoy-Like Conditions

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ^?1], and M[x, x ^?1] over suitable skew extension rings.