群环RG_p的代数性质及其结构

主要研究了当R为有限布尔环,G_p为素数p阶群时,群环RG_p的代数性质及其结构,给出了群环RG_p的单位,零因子和素谱的具体刻画,同时还给出了求解RG_p的素谱的一个一般方法.     

关于半clean群环

环$R$称为是半clean的, 是指环中的每个元素都是一个单位与一个周期元的和. clean环是半clean的. 刻画半clean群环的一般情形是不容易的. 我们的目的是考虑如下问题：若$G$ 是局部有限群或者是阶是3的循环群, 群环$RG$何时是semiclean的. clean群环上的一些已有结果被推广.     

群环Z_nG的零因子图的性质,Ⅲ

环的零因子图是20世纪90年代才兴起的一个数学研究方向.环上的零因子图的研究,刻画了环的零因子的结构,这对理解环结构本身具有重要意义.群环是群论和环论的交汇点之一.对它的研究在环论,群论及伽罗华理论等学科领域都有重要的意义.主要讨论了群环Z_nG的零因子图的性质,对群环Z_nG的零因子图的围长,平面性,直径给出了较为具体的刻画,其中G为非循环的有限交换群.     

群环Z_nG的零因子图的性质,Ⅱ

主要讨论了群环Z_nG的零因子图的性质,分别给出了群环Z_nG的零因子图的围长,直径和平面性的详细刻画,其中G为合数阶循环群.     

Smash积为超限左自由正规化扩张的充分条件及其应用

本文给出了代数A与双代数H的Smash积AH是A的超限左自由正规化扩张的一个充分条件．进一步,主要结果被运用到斜半群环、斜群环和量子群Uq（sl（2））从而给出了它们的一些性质．     

分次FS-环(英)

本文引进群分次环上分次模的分次FS－模的概念,利用分次极大分次左理想给出分次FS－环的几个刻画,得到了环R和群环RG,分次环R和分次环的群环R[G]间的几个等价条件．     

分次FS—环

本文引进群分次环上分次模的分次ＦＳ－模的概念,利用分次极大分次左理想给出分次ＦＳ－环的几个刻画,得到了环Ｒ和群环ＲＧ,分次环Ｒ和分次环的群环Ｒ「Ｇ」间的几个等价条件。     

群环Z_nG的基于理想△(G)的零因子图

主要讨论了群环Z_nG的基于理想△(G)的零因子图Γ_(△(G))(Z_nG)的性质,分别给出了Γ_(△(G))(Z_nG)的围长,平面性和直径的详细刻画.同时,给出了交换环R基于其理想I的零因子图Γ_I(R)与商环R/I的零因子图Γ(R/I)的直径的关系的一个刻画.     

有限群分次环上的余挠对研究

本文研究了余挠对在有限群分次和非分次情况下的联系.利用分次理论以及相对同调,我们首先研究了R是任意环G是有限群的情况下,余挠对在R-模范畴以及斜群环S=R*G-模范畴之间的关系;然后我们研究了R-gr范畴中刚性余挠对的等价刻画,同时给出了余挠对在R-gr范畴与R-模范畴之间的关系,其中R是G分次环,群G是有限群且|G|^-1∈R.     

有限群分次环上的余挠对研究（英文）

本文研究了余挠对在有限群分次和非分次情况下的联系.利用分次理论以及相对同调,我们首先研究了R是任意环G是有限群的情况下,余挠对在R-模范畴以及斜群环S=R*G-模范畴之间的关系;然后我们研究了R-gr范畴中刚性余挠对的等价刻画,同时给出了余挠对在R-gr范畴与R-模范畴之间的关系,其中R是G分次环,群G是有限群且|G|~(-1)∈R.     

Dubrovin Valuation Skew Group Rings

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised. Received June 8, 1999, Revised June 4, 2001, Accepted July 20, 2001     

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G _T=<1>,where G _T is the inertial group of R.     

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.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

Value Functions on Simple Artinian Rings

In this paper we introduce the notion of a value function ω on a simple Artinian ring Q whose restriction to the center of Q is a Krull valuation. We then study the structure of a valuation ring Bω corresponding to ω and its ideals. It is shown that Bω is a prime Goldie ring and the set of two-sided ideals of Bω is totally ordered by inclusion with the maximal ideal J(BW). Furthermore, it is proved that if Q is a central simple algebra, then the valuation ring Bω is a Dubrovin valuation ring which is integral over its center.     

Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

Real Closed Valuation Rings

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves.     

Completions of Convexly Ordered Valuation Rings

We prove that every convexly ordered valuation ring has a unique completion as a uniform space, which furthermore is a convexly ordered valuation ring. In addition, we give a model theoretic characterisation of complete convexly ordered valuation rings, and give a necessary and sufficient condition for the completion of a convexly ordered valuation ring to be a real closed ring. Mathematics Subject Classification : 03C60, 13L05, 54E15.     

On Skew Triangular Matrix Rings

Let α be a nonzero endomorphism of a ring R, n be a positive integer and T_n(R, α) be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by T_n(R, α). Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x; α]/(x~n), where R[x; α] is the skew polynomial ring.     

A Sufficient Condition for a Smash Product as a Transfinite Left Free Normalizing Extension and Its Application

In this paper, a sufficient condition is given under which the smash product A#H is a transfinite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group U _q (sl(2)) such that some properties are shown. Received April 14, 1997, Accepted August 10, 1998