π-凝聚环上多项式环的同调维数

本对π凝聚环上多项式环的FGT维数做了讨论，给出了定理，R，R[x]是π-凝聚环，则当脚FGT-WD(R)≥1时FGT-WD(R[x])=FGT—WD(R) 1，当FGT—WD(R)=0时，FGT-WD(R)．FGT—WD(R[x])中一为零另一个也为零．     

关于Γ-除环

文中首先给出了定理２．２,由它可以行到文献「４」和「５」中关于Γ－域和除Γ－环的一些结果,其次,我们还指出了一个Γ－环是Γ－除环的几个条件。     

有限表现维数与凝聚环

在本文中,我们从研究投射等价模的有限表现维数的关系入手,给出了有限表现维数的维数转移定理(定理2.5),并且运用有限表现维数刻划了凝聚环(定理2.4)。最后我们得到了在经典局部化下,环与模的有限表现维数的不变性定理(定理2.6,定理2.8)。     

特征模对环的刻划

设Ｒ是一个环，Ｍ是一个左Ｒ摸，Ｍ＊＝ＨｏｍＺ（Ｍ，Ｑ／Ｚ）为Ｍ的特征模．Ｒ．Ｒ．Ｃｏｌｂｙ和Ｔ．Ｊ．Ｃｈｏａｔｈａｎ等人利用特征模对ＩＦ环、凝聚环、Ｎｏｅｔｈｅｒ环、Ａｒｔｉｎ环作出了一些非常好的刻划．本文利用特征模对更为广泛的一些环作出了较有意义的刻划．     

GCD整环与自反模

本文证明了凝聚整环是ＧＣＤ整环当且仅当秩为１的自反模是自由模．同时还得到有限弱整体维数的凝聚整环是ＧＣＤ整环当且仅当Ｐｉｃ（Ｒ）＝１．特别地，有限整体维数的Ｎｏｅｔｈｅｒ整环是ＵＦＤ当且仅当Ｐｉｃ（Ｒ）＝１．     

关于凝聚局部环的正则性

本文证明了极大理想m是有限生成的交换凝聚局部环（R,M）是正则的充分必要条件是m可以由一个正则R-序列生成,推广了文献[1]中相应的结论并给出了一个由正则凝聚局部环构造大量的非正则凝聚局部环的方法．     

内同构环与内异环的结构

所有真子环都同构的结合环,称为内同构环,任两不同的子环都不同构的结合环,称为内异环.本文目的是给出内同构环与内异环的一些结构定理,从而基本上解决了Szasz F.A.提出的问题81:怎样的结合环,它的不同子环总不同构?     

t次正规环的一个恒等式定理及其应用

一个适合 t 正规多项式的环叫做 t 次正规环。本文得到一个关于 t 次正规环的恒等式定理,并在此定理基础上得出交换环的另一些恒等式,它们是著名的环的交换性问题的一些结果或类似的结果。     

右IF－环及凝聚环的挠理论

本文研究了右ＩＦ－环的性质，证明出环Ｒ是右ＩＦ－环当且仅当Ｒ是左凝聚环，并且是平坦模；由此证明出右ＩＦ－环与左ＧＱＦ－环是等价的，其次应用右ＩＦ－环研究了凝聚环的挠理论性质，证明出凝聚环与Ｔ－凝聚环的关系。     

右IF－环及凝聚环的挠理论

本文研究了右ＩＦ－环的性质，证明出环Ｒ是右ＩＦ－环当且仅当Ｒ是左凝聚环，并且是平坦模；由此证明出右ＩＦ－环与左ＧＱＦ－环是等价的，其次应用右ＩＦ－环研究了凝聚环的挠理论性质，证明出凝聚环与Ｔ－凝聚环的关系。     

On Strong n-Perfect Rings

In this article we introduce and investigate a particular class of n-perfect rings that we call “strong n-perfect rings.” We are mainly concerned with this class of rings in the context of pullbacks. We also exhibit a class of n-perfect rings that are not strong n-perfect. Finally, we establish the transfer of this notion to the direct product.     

Rigid Rings and Makar-Limanov Techniques

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

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.     

凝聚环上的Cotilting 模(英)

本文首先介绍了co－*－模的概念和刻划了凝聚环的一些性质，然后刻划了凝聚环上的Cotilting模．     

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.     

Generalized Morphic Rings and Their Applications

A ring R is called “left generalized morphic” if for every element a in R, there exists b ∈ R such that l(a)? R/Rb, where l(a) denotes the left annihilator of a in R. The aim of this article is to investigate these rings. Several examples are given. They include left morphic rings and left p.p. rings. As applications, some homological dimensions over these rings are defined and studied.     

On Rings Whose Elements are the Sum of a Unit and a Root of a Fixed Polynomial

A ring R with identity is called “clean” if for every element a ? R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)[x]. An element r ? R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ? (x ? a)(x ? b)C(R)[x] with a, b ? C(R) and b ? a ? U(R); equivalent conditions for (x² ? 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given.     

Minimalities and Modules over Dedekind-Like Rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings.     

IF环和拟ZIF环

本文定义了拟ＺＩＦ环．研究了ＩＦ环与拟ＺＩＦ环的结构．进一步地，讨论了ＩＦ环与拟ＺＩＦ环的关系，并用拟ＺＩＦ环刻划了ＩＦ环．