Ideals defining Gorenstein rings are (almost) never products

This note proves that if is an unramified regular local ring and proper ideals of height at least two, then is never Gorenstein.

     

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Generalized Stable Regular Rings

Abstract

In this paper we show that a regular ring R is a generalized stable ring if and only if for every x ∈ R, there exist a w ∈ K(R) and a group G in R such that wx ∈ G. Also we show that if R is a generalized stable regular ring, then for any A ∈ M _n (R), there exist right invertible matrices U ₁, U ₂ ∈ M _n (R) and left invertible matrices V ₁, V ₂ ∈ M _n (R) such that U ₁ V ₁ AU ₂ V ₂ = diag(e ₁,…, e _n ) for some idempotents e ₁,…, e _n  ∈ R.     

Containing Symbolic Powers in Regular Rings

We extend some results of Ein–Lazarsfeld-Smith, Hochster–Huneke, and Takagi-Yoshida concerning the containments of symbolic powers of ideals in ordinary powers in regular rings.     

NOETHERIAN GR-REGULAR RINGS ARE REGULAR

It is proved that for a left Noetherian z-graded ring A,if every finitely generated graded A-module has finite projective dimension(i.e-,A is gr-regular)then every finitely generated A-module has finite projective dimension(i.e.,A is regular).Some applications of this result to filtered rings and some classical cases are also given.     

A REMARK ON THE HENSELIZATION OF A REGULAR LOCAL RING

1.IntroductionThenotionofHenselizationofintegrallyclosedintegritydomainswasintroducedbyNagata[3]fromtheconceptofHenselianringwhichwasgivenfirstlybyAzumaya[1].TheHenselizationofalocalringAisacouple(A,i),whereAisalocalHenselianringandi:AAalocalmoaphism...     

NOETHERIAN GT-REGULAR RINGS ARE REGULAR

ＮＯＥＴＨＥＲＩＡＮＧＴ－ＲＥＧＵＬＡＲＲＩＮＧＳＡＲＥＲＥＧＵＬＡＲ￥ＬＩＨＵＩＳＨＩ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａａｎｘｉＮｏｒｍａｌＵｎｉｖｉｓｉｔｙＸｉ＇ａｎ７１００６２，Ｃｈｉｎａ．）Ａｂｓｔｒａｃｔ：Ｉｔｉｓｐｒ...     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

一类环上HX环的结构

自李洪兴１９９１年提出了ＨＸ环以来，人们一直有这么一个问题没解决，就是是否存在非平凡的ＨＸ环的例子？但至今既没找到非平凡的ＨＸ环，也没有证明任一环Ｒ仅存在平凡的ＨＸ环。针对这个问题，本文提出并证明了一类环仅有平凡ＨＸ环，还给出了一系列的结构定理。这样，既为证明任一环Ｒ仅有平凡的ＨＸ环的猜想有新的启示，也为人们指明无须在这一类环上寻找非平凡ＨＸ环。     

A Class of Auslander-Regular Macaulay Rings

Abstract

For a (left and right) noetherian semilocal ring R we analyse a regularity concept (called weak regularity) based on the equation gld R = dim R. Examples are regular Cohen-Macaulay orders over a regular local ring, localized enveloping algebras of finite dimensional Lie algebras, and the regular rings classified in Rump (2001b). We prove that weakly regular rings are Auslander-regular and Macaulay.     

正规环的Fuzzy双侧理想

本文首先提出正规环的Fuzzy双侧理想的定义,讨论了Fuzzy双侧理想和Fuzzy理想之间的关系,给出对环、Fuzzy对环、左(右)零环等概念,并在正规环中研究了它们的性质。最后在超正规环和完全正规环中研究了Fuzzy双侧理想和Fuzzy理想,得出一些有趣的结果。     

关于正则环的理想

本文把对于正则环的可比较性推广到对于正则环上理想的可比较性,得到这 种可比较性的若干刻画.进而我们证明,对稳定秩1的理想, Roth问题有肯定解答.     

准正则环与正则环

每一个主左理想均由一个幂等元生成的环叫正则环。每一个左理想均由若干个幂等元生成的环叫准正则环。本文研究准正则环与正则环的一些性质，讨论准正则环成为正则环的一些条件，准正则环、正则环与Ｖ环之间的关系，准正则环成为Ａｂｅｌ正则环的条件。     

On the Drazin index of regular elements

It is known that the existence of the group inverse a ^# of a ring element a is equivalent to the invertibility of a ² a ⁻ + 1 − aa ⁻, independently of the choice of the von Neumann inverse a ⁻ of a. In this paper, we relate the Drazin index of a to the Drazin index of a ² a ⁻ + 1 − aa ⁻. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.      

Separative Cancellation over Regular Rings

This article develops element-wise characterizations of separative ideals in regular rings. These give connections among separativity, annihilators, cokernels and unit-regularity. For strongly separativity, the analogs are also obtained.     

单因环的性质、等价条件和结构

一个有单位元的结合环，如果其元素不是单位就是零因子，则称其为单因环。它是很广泛的一个环类．本文讨论了单因环的住质、等价条件和结构；还讨论了它同其它环诸如正则环、局部环和循环环的关系，从而较深刻地刻划了这种环。     

Separative Ideals,Clean Elements,and Unit-Regularity

ABSTRACT

We prove that an ideal I of a regular ring R is separative if and only if each a ? R satisfying Rr(a)aR = Ra?(a)R = RaR(1 ? a)R ? I is unit-regular. If I is a separative ideal of a regular ring R, then each a ? R satisfying Rar(a²) = ?(a²)aR = R(a ? a²) R ? I is clean. Some applications are also obtained.