Von Neumann正则环的几点注记

本文利用理想化子的概念定义了duo环的一个推广,称为MD环,并且研究了MD环的一些性质.特别地.我们证明了:如果R是MD环,且每一个奇异单左R-模是p-内射的,那么R是指数有界的von Ncumann正则环,因此,R.Yue chi ming提出的如下公开问题得到了肯定的回答:GLD左Γ-环是否为Von ncumann正则的?     

Flatness and injectivity of simple modules over a commutative ring

In this short paper, the homological functors are used to show that a simple module is flat if and only if it is injective. It is used to give a characterization of Von Neumann regular ring, that is, a commutative ring is Von Neuman regular if and only if every simple module is injective.     

弱半局部环的同调性质

环R称为弱半局部环,如果R/J(R)是Von Neumann正则环.给出了一个交换环是弱半局部环的充分且必要条件;还讨论了交换凝聚弱半局部环及其模的同调维数.     

V-Categories: Applications to Graded Rings

Motivated by the study of V-rings, we introduce the concept of V-category, as a Grothendieck category with the property that any simple object is injective. We present basic properties of V-categories, and we study this concept in the special case of locally finitely generated categories, for instance the category R-gr of all graded left R-modules, where R is a graded ring. We use the characterizations of V-categories in the study of graded V-rings. Since V-rings are closely related to Von Neumann regular rings (in the commutative case these classes of rings coincide), the last part of the article is devoted to graded regular rings.     

PRIME AND SEMIPRIME BI-IDEALS

Abstract

Prime and semiprime bi-ideals in associative rings are defined. This provides a setting for a generalization of the well-known theorem that a commutative ring is Von Neumann regular iff every ideal is semiprime.     

Cartan-Eilenberg projective,injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.     

von Neumann Regular Rings and Right SF-rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this paper, we study the regularity of right SF-rings and prove that if R is a right SF-ring whose all maximal (essential) right ideals are GW-ideals, then R is regular.     

von Neumann正则环与右SF-环

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this paper, we study the regularity of right SF-rings and prove that if R is a right SF-ring whose all maximal (essential) right ideals are GW-ideals, then R is regular.     

On (Strongly) Gorenstein Von Neumann Regular Rings

In this article, we introduce and study the rings over which every module is (strongly) Gorenstein flat, which we call them (strongly) Gorenstein Von Neumann regular rings.     

Invariant subrings of separable algebras

This paper gives a necessary and sufficient condition that the ring of invariants of every group of automorphisms of every projective, separable, commutative algebra over a given commutative ring is itself a union of separable, projective subalgebras. Rings satisfying the condition include products of connected rings, von Neumann regular rings, and some rings of functions.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

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。     

Simple flat Leavitt path algebras are von Neumann regular

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40?years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras.     

Left SF-Rings and Regular Rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

RD-Flatness and RD-Injectivity

It is proven that every commutative ring whose RD-injective modules are Σ-RD- injective is the product of a pure semisimple ring and a finite ring. A complete characterization of commutative rings for which each Artinian (respectively simple) module is RD-injective, is given. These rescan be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively the RD-relativization of flat modules and fp-injective modules. It is also shown that a commutative ring is perfect if and only if each RD-flat module is RD-projective.     

Entropy in the category of perfect complexes with cohomology of finite length

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic.Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is “additive”. Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.     

von-Neumann正则环与左SF-环

环R称为左SF-环,如果每个单左R-模是平坦的.众所周知,Von-Neumann正则环是SF-环,但SF-环是否是正则环至今仍是公开问题,本文主要研究左SF-环是正则环的条件,证明了:如果R是左SF-环且R的每个极大左(右)理想是广义弱理想,那么R是强正则环.并且推广了Rege[3]中的相应结果.     

n- 强Gorenstein FP- 内射模

设n 是正整数, 本文引入并研究n- 强Gorenstein FP- 内射模. 对于正整数n > m, 给出例子说明n- 强Gorenstein FP- 内射模未必是m- 强Gorenstein FP- 内射的, 并讨论n- 强Gorenstein FP-内射模的诸多性质. 最后, 利用n- 强Gorenstein FP- 内射模刻画n- 强Gorenstein Von Neumann 正则环.     

On free commutative regular rings

It has been observed that the category of all regular rings, when viewed as a full subcategory of the category of all rings, is not an algebraic variety. However if a regular ring is viewed as a ring equipped with a unary operation q such that xx^qx = x and 0000^q= 0, then the category of all rings with this added structure is indeed a variety but it is not, in any natural way, a subcategory of the category of all rings. In this article regular rings are viewed in this way and the free commutative regular rings are constructed. They are derived from the universal commutative regular ring associated with certain polynomial rings.