McCoy环的扩张(英文)

A ring R is said to be right McCoy if the equation f(x)g(x)=0,where f(x)and g(x)are nonzero polynomials of R[x],implies that there exists nonzero s∈R such that f(x)s=0.It is proven that no proper(triangular)matrix ring is one-sided McCoy.It is shown that for many polynomial extensions,a ring R is right McCoy if and only if the polynomial extension over R is right McCoy.     

广义线性McCoy环

称环R是右线性McCoy的,如果R[x]中非零线性多项式f（x）,g（x）满足I（x）g（x）=0,则存在非零元素r∈R使得f（x）r=0．设a是环R的自同态,通过用斜多项式环R[x;a]中的元素代替一般多项式环R[x]中的元素而引入a-线性McCoy环的概念．讨论了a-线性McCoy环的基本性质和扩张性质．     

PM环的扩张

A ring is called PM-ring if every primitive ideal is a intersection of its some maximal ideals.In this paper,the property of PM-ring is discussed and the necessary and sufficient condition are given for the polynomial,centralizing,strongly normal extensions of PM-ring to be PM-rings.We also get some applications of the extension of PM-ring in group ring.     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

M-McCoy环和M-Armendariz环的多项式扩张

研究非交换环上的相对于幺半群的McCoy环和Armendariz环的多项式扩张.对于包含无限循环子幺半群的交换可消幺半群M,证明了若R是M-McCoy（或M-Armendariz）环,则R上的洛朗多项式环R[x,x-1]是M-McCoy（或M-Armendariz）环.     

二次截面Nakayama代数的McCoy性

Zhou Yuye;Cheng Zhi(School of Mathematics and Statistics,Anhui Normal University,Wuhu 241003,China)     

Baer环和拟-Baer环的多项式扩张的一点注记

I1和I2分别是环R的一个左理想和右理想,T1=R[x]和T2=R[x,x-1]分别表示多项式环和洛朗多项式环.首先给出两个例子,分别说明了T1I1不一定是T1的左理想与T2L2不一定是T2的右理想.其次给出了环的多项式扩张及洛朗扩张的理想的性质.最后证明了,若R[X](R[x,x-1])是拟-Baer环,则R也是拟-...     

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．     

关于全不变扩张环和模(英文)

In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.     

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.     

The McCoy Condition on Ore Extensions

Nielsen [29 Nielsen , P. P. ( 2006 ). Semi-commutativity and the McCoy condition . J. Algebra 298 : 134 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.     

Linearly McCoy Rings and Their Generalizations

A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.     

Ore Extensions of Skew Armendariz Rings

Let α be an endomorphism and δ an α-derivation of a ring R. We introduce the notion of skew-Armendariz rings which are a generalization of α-skew Armendariz rings and α-rigid rings and extend the classes of non reduced skew-Armendariz rings. Some properties of this generalization are established, and connections of properties of a skew-Armendariz ring R with those of the Ore extension R[x; α, δ] are investigated. As a consequence we extend and unify several known results related to Armendariz rings.     

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.