Characterizations of Strongly Regular Rings

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Strongly Regular Rings and SF-rings

in this paper, new characteristic properties of strongly regular rings are' given.Relations between certain generalizations of duo rings are also considered. The followingconditions are shown to be equivalent: (1) R is a strongly regular ring; (2) R is a left SFring such that every product of two independent closed left ideals of R is zero; (3) R is aright SF-ring such that every product of two independent closed left ideals of R is zero; (4)R is a left SF-ring whose every special left annihilator is a quasi-ideal; (5) R is a right SFring whose every special left annihilator is a quasi-ideal; (6) R is a left SF-ring whose everymaximal left ideal is a quasi-ideal; (7) R is a right SF-ring whose every maximal left ideal isa quasi-ideal; (8) R is a left SF-ring such that the set N(R) of all nilpotent elements of R isa quasi-ideal; (9) R is a right SF-ring such that N(R) is a quasi-ideal.     

环的（几乎）优越扩张

设环S是环R的优越扩张．本文证明了如一环是右IF-环；则另一环亦是，同时还得出了一个S是SF-环是正则的充要条件．     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

Excellent Extensions of Rings

Ｅｘｃｅｌｌｅｎｔ　Ｅｘｔｅｎｓｉｏｎｓ　ｏｆ　ＲｉｎｇｓＬｉｕＺｈｏｎｇｋｕｉ（刘仲奎）ａｎｄＷａｎｇＴｉｎｇＺｈｅｎ（王廷桢）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｃｍａｔｉｃｓ，ＮｏｒｔｈｗｅｓｔＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｌａｎｚｈｏｎ，７...     

广义幂级数环的PP性质

作为幂级数环的推广,Ｒｉｂｅｎｂｏｉｍ引入了广义幂级数环的概念．设Ｒ是有单位元的交换环,（Ｊ,≤）是严格全序半群．本文中我们证明了如下结果：（１）广义幂级数环 ［［Ｒｓ］］是ＰＰ－环当且仅当Ｒ是ＰＰ－环且Ｂ（Ｒ）的任意 Ｓ－可标子集Ｃ在Ｂ（Ｒ）中有最小上界;（２）如果对任意ｓ∈Ｓ都有０≤ｓ,则［［Ｒｓ,≤］］是弱ＰＰ－环当且仅当Ｒ是弱ＰＰ－环．我们还给出了一个例子说明交换的弱ＰＰ－环可以不是ＰＰ－环．     

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.     

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?     

替换环什么时候是冯诺依曼正则环（英文）

We study when exchange rings are von Neumann regular. An exchange ring R with primitive factors Artinian is von Neumann regular, if the Jacobson radical of any indecomposable homomorphic image of R is T-nilpotent, and if any indecomposable homomorphic image of R is semiprime. Every indecomposable semiprimitive factor ring of R is regular, if R is an exchange ring such that every left primitive factor ring of R is a ring of index at most n and if R has nil-property.     

On G-Regular Local Rings

In this article, we define a G-regular local ring as a commutative, noetherian, local ring, over which all totally reflexive modules are free. We study G-regular local rings and observe that they behave similarly to regular local rings. We extend Eisenbud's matrix factorization theorem and Knörrer's periodicity theorem to G-regular local rings.     

The structure of commutative rings with monomorphic flat envelopes

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose structure is given in terms of OF rings. It turns out that a commutative ring R with m.f.e. is characterized as a (essential) subdirect product of a locally QF ring with m.f.e. and a Von Neumann regular ring, with the latter flat as an R-module.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

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．     

分次Abel正则环的结构

首先给出了 gr-正则环为分次除环的两个充要条件 ,其次讨论了分次正则环r G(R)和分次 Jacobson根 JG(R)之间的关系 ,最后给出了分次 Abel正则环的结构定理 .     

关于AP-内射环的一个注记

本文的主要目的是讨论AP-内射环中的两个问题:(1)环R是正则的当且仅当R是左AP-内射的左PP-环;(2)如果R是左AP-内射环,那么R是内射环当且仅当R是弱内射环.因此我们推广了内射环的一些结果,与此同时我们还取得了一些新的结果.     

HX环的同态

文[1]首先建立了HX环的概念，将环上的代数结构自然地引导出它的幂集上的某种代数结构。类似的问题是：环的同态能否诱导出它们的HX环之间的同态?本文较详细地讨论HX环的同态。     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

ON SOME PROPERTIES OF IF RINGS

Abstract

We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.     

自内射环的余Hopf性

本文证明了自内射环R是余Hopf的当且仅当R满足stablerangeone.于是得到了Varadarajan在[9]中的公开总是对于自内射环是成立的,即M_n(R)是余Hopf的当且仅当R是余Hopf的.作为应用证明了Goodeal的一个公开问题对于自内射正则环有肯定的回答.     

正则模与半单Artin环

本文用则模的术语给出了半单Artin 环的刻划。得到如下三个条件的等价性:(1)R 是一个半单Artin 环;(2)每一个R-模都是正则模;(3)每一个单纯R-模都是正则模。