域上F_4型Chevalley群中含单项子群的极大子群

对特征不为 ２的任意域上 Ｆ_４型 Ｃｈｅｖａｌｌｅｙ群,构造了一类真包含单项子群的子群,从而否定回答 了单项子群的极大性问题,同时证明了所构造的群恰为极大子群     

域上 F4型 Chevalley 群中含单项子群的极大子群

对特征不为2的任意域上F4型Chevalley群,构造了一类真包含单项子群的子群,从而否定回答了单项子群的极大性问题.同时证明了所构造的群恰为极大子群.     

域上Cl型Chevalley群中含单项子群的极大子群

对特征不为2的任意域上Cl型Chevalley群.构造了一类真包含单项子群的子群,从而否定回答了单项子群的极大性问题,同时证明了所构造的群恰为极大子群.     

置换性很好的一类群

主要研究子群置换性质对有限群结构的影响.通过子群的置换性得到一类群,即B 群.B群是全可置换群的扩展,利用全可置换群的p次中心扩张和子群的阶得到B群的一些性质并对B 群的结构进行一些刻画.应用B 群的结构得到有限p （p〉2）群为二元生成的B 群的充要条件.     

域上Cl型Chevalley群中含单项子群的极大子群

对特征不为2的任意域上Ci型Chevaley群,构造了一类真包含单项子群的子群,从而否定回答子单项子群的极大性问题,同时证明了所构造的群恰为极大子群。     

无限正则p-群

对一类无限正则p-进行了研究,得到了一个正则的局部幂零P-群G如果满足|G:(?)_1(G)|<∞,那么G是幂零的且G是可除阿贝尔P-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.     

极小非MSSP-群的分类

本文研究了一类有限极小非∑-群.利用群G的每个Sylow子群的极小子群的s-半置换性对G的结构影响,获得了一类新的极小非∑-群的分类,推广了极小非Σ-群部分研究成果.     

Weyl群的子群结构

对各种类型的不可约根系,讨论了其Ｗｅｙｌ群一类子群的极大性。     

无限正则p-群

对一类无限正则p-群进行了研究,得到了一个正则的局部幂零p-群G如果满足|G(Ω)1(G)|＜∞,那么G是幂零的且G是可除阿贝尔p-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.     

一类pnm阶群的构造

本文研究了一类特殊的pnm阶有限群的构造.利用求解数论同余方程的方法和群的扩张理论,得到了具有m阶循环正规子群,其补子群为循环群的Pnm阶有限群的构造及相关的计数定理.     

SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.     

MDS and self-dual codes over rings

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings. We also define the Reed–Solomon codes over principal ideal rings.     

Independence of vectors in codes over rings

We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors. Hongwei Liu is supported by the National Natural Science Foundation of China (10571067).     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

Counting codes over rings

We extend the definition of free codes to codes over local rings and arbitrary Frobenius rings. The number of free codes over finite Frobenius rings is determined by calculating the number for local rings and applying the Chinese Remainder Theorem. A formula for the number of codes of arbitrary type over a finite chain ring is given and this is applied to determine the number of linear codes over a finite principal ideal ring.     

When is Every Module with Essential Socle a Direct Sums of Quasi-Injectives?

In the present article we study the structure of rings, over which essential extensions of semisimple modules are direct sums of quasi-injectives. In the special case of commutative rings, these rings are precisely Artinian PIR and so every module over such rings is a direct sum of cyclics as characterized by Köthe and Cohen-Kaplansky.     

Intermediate rings between matrix rings and Ornstein dual pairs

We show that isomorphism of intermediate rings between row and column finite matrix rings and row finite matrix rings implies Morita equivalence of the coefficient rings and equality of the cardinality of the set of indices. Among the applications we extend the Isomorphism Theorem for Dual Pairs over Division Rings to Ornstein dual pairs over any class of rings for which Morita equivalence implies isomorphism.     

Noncommutative biorthogonal polynomials

We define and study biorthogonal sequences of polynomials over noncommutative rings, generalizing previous treatments of biorthogonal polynomials over commutative rings and of orthogonal polynomials over noncommutative rings. We extend known recurrence relations for specific cases of biorthogonal polynomials and prove a general version of Favard?s theorem.     

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)