Excellent扩张和弱维数

在这篇文章里，我们证明了，当环Ｓ是Ｒ的ｅｘｃｅｌｌｅｎｔ扩张，Ｍ是Ｓ－模时，Ｍ做为Ｓ－模的弱维数与Ｍ做为Ｒ－模的弱维数相等。     

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．     

环的Excellent扩张

设S是R的Excellent扩张.本文讨论了R的性质对S的影响及S的性质对R的影响,并利用所得结果改进和推广了一些已知的结论.     

关于完全环与Noether环

（ｉ）环Ｒ是左完全环，当且仅当存在一个基数ｃ，使得任意平坦左Ｒ－模是一个拟投射模和一个ｃ－限制的ＥＳ－模的直和。（ｉｉ）Ｒ是左Ｎｏｅｔｈｅｒ环，当且仅当存在一个基数ｃ，使得任意内射左Ｒ－模的直和是一个（拟）连续模和一个ｃ－限制的ＥＳ－模的直和。     

Exchange环及其比较条件

该文得到了Ｅｘｃｈａｎｇｅ性质的内直和刻画,给出了局部环的一个Ｅｘｃｈａｎｇｅ新特征,进一步地研究了Ｅｘｃｈａｎｇｅ环上相关比较结构．     

分次环的分次Excellent扩张

本文引进了分次环的分次Excellent扩张概念,设S=⊕_(g∈G)S_g是R=⊕_(g∈G)R_g的分次Excellent扩张,证明了S是分次右V-环当且仅当R是分次右V-环,S是分次PS-环当且仅当R是分次PS-环,S是分次Von Neumann正则环当且仅当R是分次Von Neumann正则环。     

关于基本可数生成子模的扩张性

称左R-模M是ecg-扩张模,如果M的任意基本可数生成子模是M的直和因子的基本子模.在研究了ecg-扩张模的基本性质的基础上,本文证明了对于非奇异环R,所有左R-模是ecg-扩张模当且仅当所有左R-模是扩张模.同时我们还用ecg-拟连续模刻画了Noether环和Artin半单环.     

泛代数的二次扩张

在本文中引入了泛代数的二次扩张的概念，解决了ＴＵ－ＥＣ（Ａ）和ＵＴ－ＥＣ（Ａ）的存在、真类和基数问题，范畴ＴＵ－ＥＣ（Ａ）（及ＵＴ－ＥＣ（Ａ）），并得到了有关二次扩张的几个同构定理，还对一点二次扩张作了讨论。     

带许多单位正则元的环

提要 本文研究带许多单位正则元的环,给出这类环的刻划．特别地,证明了这类环都是拟泛ＧＥ２环作为应用,还证明本原分式为 Ａｒｔｉｎ的 Ｅｘｃｈａｎｇｅ环是拟泛 ＧＥ２环．     

扩张矩阵的一些性质

研究了相关于扩张矩阵A的扩张球和拟范数的一些性质.首先通过具体实例及欧氏范数关于A的上下界估计指出扩张矩阵与经典球及欧氏范数匹配不佳,但欧氏范数相关于A仍能保持全局伸缩性.其次研究了相适应于扩张矩阵的扩张球和拟范数关于伸缩性、凸性、可积性、微分估计及傅里叶变换的一些性质.最后通过欧氏范数与相关于扩张矩阵的拟范数的不等式估计证明了相关于拟范数的两类施瓦茨函数空间和相关于欧氏范数的经典施瓦茨函数空间都是等价的.     

Finite Normalizing Extensions of FSN Rings

Fully semiprimary Noetherian bimodules and their bimodule extensions are examined. In the presence of incomparability of the link graph of prime ideals, certain bimodule extensions preserve the fully semiprimary property. In particular, a finite normalizing extension ring of a fully semiprimary Noetherian ring is also fully semiprimary as a bimodule over the base ring. It is shown that the extension ring is itself a fully semiprimary ring. An application to crossed products over finite groups is given.     

On the Extension of PM-Ring

ＯｎｔｈｅＥｘｔｅｎｓｉｏｎｏｆＰＭ－Ｒｉｎｇ朱元森ＯｎｔｈｅＥｘｔｅｎｓｉｏｎｏｆＰＭ－Ｒｉｎｇ￥ＺｈｕＹｕａｎｓｅｎ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＨｅｂｅｉＴｅａｃｈｅｒｓＵｎｉｖ．，Ｓｈｉｊｉａｘｈｕａｎｇ，０５００１６）Ａｂｓｔｒａｃｔ：Ａｒｉ...     

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     

LINEAR ORDERS ON RINGS

The general ideas introduced in Radeleczki and Szigeti (2004) are adapted to investigate quasi cones and cones of rings. Using the finite extension property for cones, we answer the question concerning when a compatible partial order of a ring has a compatible linear extension (equivalently, when the positive cone of this order is contained in a full cone). It turns out that, if there is no such extension, then it is caused by a finite system of polynomial-like equations satisfied by some elements of a certain finite subset of the ring and some positive elements.     

物元可拓集集合性质研究

在可拓集合概念基础上,提出了向量可拓集和区间可拓集概念,研究了物元可拓集合的交、并、求补等集合运算及其性质.     

Gorenstein平坦模与Frobenius扩张

设R■A是环的Frobenius扩张,其中A是右凝聚环,M是任意左A-模.首先证明了_AM是Gorenstein平坦模当且仅当M作为左R-模也是Gorenstein平坦模.其次,证明了Nakayama和Tsuzuku关于平坦维数沿着Frobenius扩张的传递性定理的"Gorenstein版本":若_AM具有有限Gorenstein平坦维数,则Gfd_A(M)=Gfd_R(M).此外,证明了若R■S是可分Frobenius扩张,则任意A-模(不一定具有有限Gorenstein平坦维数),其Gorenstein平坦维数沿着该环扩张是不变的.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Flat modules and the behavior of a standard basis relative to an extension of the ground ring

The property of a generating set of a polynomial ideal or of an ideal of a free associative algebra over a commutative ring to be its Gr?bner basis is kept by a flat (but not any) extension of the ground ring. The converse is proved in the broader context of standard bases for filtered modules.     

关于可拓集合的运算

根据蔡文提出的可拓集合的新定义,给出了可拓集合的包含、并、交、非运算的新定义,并讨论了有关运算性质,进而获得可拓域与稳定域的几个交并运算结果.