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.     

右对称环

本文在左对称环的基础上提出了右对称环的概念,分别给出了是右对称环但不是左对称环和是左对称环但不是右对称环的例子．证明了（1）如果R是Armendariz环,则R是右对称环的充要条件R[x]是右对称环;（2）如果R是约化环,则R[x]/（x^n）是右对称环,其中（xn）是由xn生成的理想．     

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.     

Polynomial Rings over Non-noetherian Power Series Rings

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Symmetric Elements Under Oriented Involutions in Group Rings

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ? of the group G, an oriented invoultion ^σ? of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.     

Group Rings Whose Symmetric Units are Solvable

Let K be an infinite field of characteristic different from 2, and G a group. Under suitable restrictions upon G, we classify the groups such that the symmetric units of KG satisfy the solvability identity (x ₁, x ₂,…, x _{2 ⁿ} ) ^o  = 1, for some n.     

The McCoy Condition on Skew Polynomial Rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ? R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

交换环上多项式结式的性质

本文主要讨论交换环上多项式结式的一些性质.首先,我们证明了交换环上一种乘积的结式等于结式的乘积的性质,然后,我们证明了交换环上一种和的结式具有的性质,并且给出了交换环上结式为零的一个充分条件.     

Group Rings Whose Symmetric Units Generate an n-Engel Group

Let F be an infinite field of characteristic different from 2 and G a torsion group. Write 𝒰⁺(FG) for the set of units in the group ring FG that are symmetric with respect to the classical involution induced from the map g ? g ^?1, for all g ∈ G. We classify the groups such that ?𝒰⁺(FG)? is n-Engel.     

On Unimodular Rows over Polynomial Rings

Let A be a commutative ring and n 3 a positive integer. In this paper, we consider unimodular rows (f₁(x),..., f_n(x)) over A[x]. We prove that, if the row of the leading coefficients of f_i(x) is unimodular over A and a A, then there exists E_n(A[x]) such that (f₁(x),...,f_n(x)) = (f₁(a),...,f_n(a)). Also, if A is a Noetherian ring with finite Krull dimension and the row of leading coefficients satisfies the same condition, then we give a bound for the length of in terms of elementary transvections.Partially supported by INTAS 93–436EXT and DFG–RFBR grant No. 96–01–00092G.2000 Mathematics Subject Classification: 19A13, 19B14, 13C10, 13B25, 13F20     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

Symmetric Bilinear Forms over Valuation Rings

本文讨论赋值环上的对称线性型、二次型和对称矩阵的合同标准形。     

Z／（p^e）上分裂环的结构

本文研究了环ｚ／（ｐｅ）上多项式的性质和分裂环的结构．主要分析了分裂环中元素的极小多项式，零化理想的结构，和分裂环子环性质．     

对称环的扩张

本文首先考虑了对称环的性质和基本的扩张．其次讨论了几种多项式环的对称性,且证明了:如果R是约化环,则R[x]／(xn)是对称环,其中(xn)是由xn生成的理想,n是一个正整数．最后证明了:对一个右Ore环R,R是对称环当且仅当R的古典右商环Q是对称环．     

具有对称自同态的环

通过引入对称α-环的概念,拓广对称环的研究.讨论对称α-环与相关环的关系,给出对称α-环的一些扩张性质,证明了1)设α是约化环R的自同态且α-1)=1.如果R是对称α-环,则R[x]/〈x~n〉是对称α-环;2)设α是右Ore环R的自同构,Q(R)是R的典范右商环.如果R是对称环,则R是对称α-环当且仅当Q(R)是对称α-环.     

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.     

Partial Skew Polynomial Rings and Jacobson Rings

In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.