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.     

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。     

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。     

几类正则环的结构

本文推广文「２」「３」「４」「５」中的结果，一方面给出了正则环为除环的两个条件，另一方面给出了两类正则环的结构定理。     

正则环上矩阵分解

利用幂等矩阵和单边可逆阵,给出了正则环上具有群逆的矩阵结构,并证明了具有奇数特征的单边单位正则环上的矩阵都可分解为两个单边可逆矩阵和形式。     

Characterizing Local Rings via Test Modules

This article shows some instances where properties of a local ring are closely connected with the homological properties of a single module.     

GROUP ACTIONS ON VON NEUMANN REGULAR RINGS

ＧＲＯＵＰＡＣＴＩＯＮＳＯＮＶＯＮＮＥＵＭＡＮＮＲＥＧＵＬＡＲＲＩＮＧＳ￥ＺＨＡＮＧＹＩＮＨＵＯＡｂｓｔｒａｃｔ：ＬｅｔＡｂｅａｒｉｎｇｗｉｔｈｉｎｄｅｎｔｉｔｙ，ＧａｆｉｎｉｔｅｇｒｏｕｐｏｆａｕｔｏｍｏｒｐｈｉｓｍｓｏｆＡ．Ｔｈｅｍａｉｎｒｅｓｕ...     

A REMARK ON THE HENSELIZATION OF A REGULAR LOCAL RING

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

分次Armendariz环与P.P.环

引进分次Armendariz环的概念,讨论了分次环R= n∈2Rn及由它导出的非分次环R,R0,及R[x]之间关于Armendariz环性质的关系,并推广了[8]的结论,得到在R= n∈ZRn是Z-型正分次环的前提下,若R是分次Armendariz,分次正规环,则R是P．P环（Bear环）当且仅当R是分次P．P．环（分环Baer环）。     

NOTE ON REGULAR D—OPTIMAL MATRICES

Let A be a j x d (0,1) matrix. It is known that if j = 2k - 1 is odd, then det(AAT) ≤ (j+1)((j+1)d/4j)j; if j is even, then det(AAT) ≤ (j+1)((j+2)d/4(j+1))j. A is called a regular D-optimal matrix if it satisfies the equality of the above bounds. In this note, it is proved that if j = 2k - 1 is odd, then A is a regular D-optimal matrix if and only if A is the adjacent matrix of a (2k - 1, k, (j + l)d/4j)-BIBD; if j = 2k is even, then A is a regular D-optimal matrix if and only if A can be obtained from the adjacent matrix B of a (2k + 1,k + 1,(j + 2)d/4(j +1))-BIBD by deleting any one row from B. Three 21 x 42 regular D-optimal matrices, which were unknown in [11], are also provided.     

REMARKS ON QUASI-PERFECT RINGS AND FC-RINGS

设Ｒ是有单位元的环，Ｓ是Ｒ的几乎优越扩张，Ｇ是有限群且｜Ｇ｜－１∈Ｒ．证明了Ｒ是ＦＣ－环（拟完备环，凝聚环）当且仅当Ｓ是ＦＣ－环（拟完备环，凝聚环），也当且仅当Ｓｍａｃｈ积Ｒ＃Ｇ＊是ＦＣ－环（拟完备环，凝聚环）．     

K-THEORETICALLY COMPLETE VON NEUMANN REGULAR RINGS

The authors introduce the concept of Ko-complete ring and study the properties of Ko-complete unit-regular rings.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

THE P-COTORSION DIMENSIONS OF MODULES AND RINGS

Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion dimension and other homological dimensions. This dimension has nice properties when the ring in consideration is generalized morphic.     

环与剩余类环的同调维数

Ｓａｎｄｏｍｉｅｒｓｋｉ Ｆ．Ｌ,Ｓｍａｌｌ Ｌ．Ｗ,和 Ｆｉｅｌｄｓ Ｋ．Ｌ．［１－２］在“幂零”条件下研究了环与约化环的同调维数．然而对一些环（如交换 Ｖｏｎ Ｎｅｕｍａｎｎ正则环）,“幂零’的条件是不成立的．因此,在本文中我们考虑非“幂零”条件下（如Ｒ（Ｒ／Ｉ）（（Ｒ／Ｉ）Ｒ）是Ｒ－投身的或Ｒ（Ｒ／Ｉ）Ｒ是Ｒ－平坦的）,环与约化环的同调维数．     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

A Note on the Graded K-Theory of Certain Graded Rings

Following ideas of Quillen we prove that the graded K-theory of a ? ⁿ -graded ring with support contained in a pointed cone is entirely determined by the K-theory of the subring of degree 0 elements.