The Embedding of Radical Rings in Simple Radical Rings

In the first correction, induction is not needed (and in factcannot be used). Thus the sentences "We shall use inductionon n,", "by the induction hypothesis" and "The proof still appliesfor n = 1" on lines 9, 7 and 3 from the end, should be omitted.     

分次环的分次根

给出了建立分次环根的一般方法．作为其应用，建立了分次环的分次Brown—McCoy根，并给出了Brown—McCoy半单分次环的结构定理．     

环的根性质

研究的对象是不含单位元分次环 R,首先证明了有关 Jacobson根的重要性质 J( R) =J( S)∩ R,其中 S =R× Z.然后利用它得到了一些好的性质     

Embedding Ordered Valued Domains into Division Rings

We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings. The author acknowledges the financial support from the state budget by the Slovenian Research Agency (project no. Z1-9570-0101-06).     

Supersemiprime Rings and Supersemiprime Radical

§1.　IntroductionInthisrecentyears,theprimeringswithspecialpropertieshavebeenstudiedsomeau-thors,forexample,N.T.Groenewaldintroducedthecompletelyprimeringsandthecompletelyrimeradicalin[1].D.HandelmanandJ.Lawrencediscussedthestronglyprimeringsandthestronglyprimeradicalin[2].S.Veldsmanstudiedthesuperprimeringsandthesuperprimeradicalin[3].ZhangXianjunintroducedtheAbsolutelysemiprimeringsandtheabsolutelysemiprimeradicalandsoon.Theaimofthispaperistostudythesemiprimeringswithspecialpropertiesan…     

Radical Semigroup Rings and the Thue--Morse Semigroup

Let R be an associative ring with unit, let S be a semigroup with zero, and let RS be a contracted semigroup ring. It is proved that if RS is radical in the sense of Jacobson and if the element 1 has infinite additive order, then S is a locally finite nilsemigroup. Further, for any semigroup S, there is a semigroup T S such that the ring RT is radical in the Brown--McCoy sense. Let S be the semigroup of subwords of the sequence abbabaabbaababbab..., and let F be the two-element field. Then the ring FS is radical in the Brown--McCoy sense and semisimple in the Jacobson sense.     

On the Factoriality of Local Rings of Small Embedding Codimension

Let A be a local ring of dimension d. If A is a quotient of a regular local ring of dimension n = d+r, then we say that A has embedding codimension ≤ r. This paper investigates some special properties of local rings of small embedding codimension. The main idea is to exploit a result in [17], which says that local rings of small embedding codimension and depth ≥ 3 are parafactorial. This tells us, with suitable additional hypotheses, that the ring is factorial, or Gorenstein, or even a complete intersection.     

群分次环的分次Levitzki根

建立了分次环的分次Levitzki根,并给出了分次Levitzki半单环的结构定理及相关结果.     

分次环的分次Jacobson根

本文通过引入弱拟正则元的概念，对一般Ｍｏｎｏｉｄ分次环Ａ（未必有１）给出以内部元素刻划的分次Ｊａｃｏｂｓｏｎ根ＪＧ（Ａ）．证明当Ａ有１时，ＪＧ（Ａ）与通常定义的Ｊｇ（Ａ）相等．对ＪＧ（Ａ）性质的讨论，推广了最近的许多结果．作为应用，我们给出了Ａｒｔｉｎ分次环的全部基本结构定理．     

Local Rings of Embedding Codepth 3. Examples

A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth 3. We give examples of algebra structures that have been conjectured not to occur.     

Serial Rings with T-Nilpotent Prime Radical

In this paper we consider serial rings with T-nilpotent prime radical, factor-rings of which by the prime radical are right Noetherian rings. We prove that the prime quiver of such a ring is a disconnected union of cycles and chains. In the case when the prime quiver of such a serial ring is a chain the prime radical is nilpotent. For serial rings with nilpotent prime radical we introduce an analogue of Kupisch series. Presented by Yu. Drozd Mathematics Subject Classifications (2000) 16P40, 16G10.     

Radical Rings with Soluble Adjoint Groups

An associative ring R, not necessarily with an identity, is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R^∘ under the circle operation r ∘ s = r + s + rs on R. It is proved that every radical ring R whose adjoint group R^∘ is soluble must be Lie-soluble. Moreover, if the commutator factor group of R^∘ has finite torsion-free rank, then R is locally nilpotent.     

关于 Radicaltotal 环上的投射模

文中给出了Radicaltotal环上投射模的分解定理. 对一个 ~uniform 维数有限的 Totalfree 环 R, 该文证明 R 是一个总体维数≤ 1 的诺特环, 且 R上的任何投射模必同构于$ \bigoplus\limits_{i\in I}Re_{i}$, 其中每个 $e_{i}$ 均为 $R$ 的非零幂等元. 此外, 文中还给出了一些相关的例子.     

有限结合环类中的特殊根

设Ω是有限结合环类中全部弱单环组成的环类，Ω１∪Ω２＝Ω，Ω１∩Ω２＝Φ，在有限结合环类中，我们证明了ＬΩ１＝ＵΩ２可以成立，并给出等式成立的充要条件．使用这个结论，我们可以证明，在有限结合环类中，超幂零根是特殊根．     

The Radical Formula for Noncommutative Rings

Ukrainian Mathematical Journal - We determine some classes of left modules satisfying the radical formula in a noncommutative ring. We also show that, under a certain condition, a finitely...     

格序环的一个根的结构

从不同角度刻画了格序环Ｒ的Ｐ－根和ｌ－Ｂ根，并对ｌ－Ｑ根环进行了讨论．揭示了Ｒ及Ｒ上的全矩阵环Ｒｎ的ｌ－Ｑ根，ｌ－Ｑ理想，素ｌ－理想，半素ｌ－理想之间的关系．     

On Sums of Radical and Regular Rings

We find conditions which ensure that a ring is adjoint regular provided that it is a sum of a radical subring with an adjoint regular subring. We also provide a criterion of adjoint regularity for a ring which is a sum of its radical and a regular subring.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 71–75, 2003.     

The l-quasi Nil Radical of Lattice-ordered Rings

§ 1.l- q Ideals and the l- q Radical of an l- ring　 Generalizing the notion of quasi nil of rings to lattice-ordered rings(l-rings) ,we de-fine l-quasi nil ideals(l-q ideals) and the l-quasi nil radical(l-q radical) of l-rings.Theproperties of the l-q radical and the structure of l-q semisimple rings are studied.Fur-thermore,we consider the l-q radical of full l-matrix l-rings and l-rings with minimumcondition on single sided l-ideals.　 Throughout this paper R denotes an l-ring if not spec…     

On Subrings of Simple Artinian Rings

For an n × n matrix ring over a division ring we investigate the structure of all subrings which have the property that every projection onto a component is either the zero map or surjective. It turns out that many of these rings are semiprimary.