Topologically Irreducible Representations and Radicals in Banach Algebras

It is shown that the topologically irreducible representationsof a normed algebra define a certain topological radical inthe same way that the strictly irreducible representations definethe Jacobson radical and that this radical can be strictly smallerthan the Jacobson radical. An abstract theory of ‘topologicalradicals’ in topological algebras is developed and usedto relate this radical to the Baer radical (prime radical).The relations with topologically transitive representationsand standard representations in the sense of Meyer are alsoexplored. 1991 Mathematics Subject Classification: 46H15, 46H25,16Nxx.     

HOPF-JACOBSON RADICAL FOR COMODULE ALGEBRAS

51.IntroductionFisher[7]discussedtheHopf-JacobsonradicalJH(A)oftheH--modulealgebraA,whereHisanirreducibleHopfalgebra.In[5],CatgeneralizedFisher'sresult.HeprovedthatJH(A)#HgJ(A#H),whereHisanarbitraryHopfalgebra,AanH--modulealgebra.AversionoftheChevalley-JacobsondensitytheoremforH-modulealgebrawasalsoprovedbyCans].Dually,LiuGuilong[8]definedandstudiedtheHopf-JacobsonradicalofH-comodulealgebraA.Inthispaper,westudyfurthertheHopf-JacobsonradicaloftheH--comodulealgebraA,andwegiveaver…     

The Baer and Jacobson Radicals of Crossed Products

The relation between the H-radical of R and radical of R#H is obtained. Using radical theory, the author shows that if H is a finite-dimensional semisimple, cosemisimple, and either commutative or cocommutative Hopf algebra, then R is H-semiprime iff R is semiprime iff R#H is semiprime.     

Strong-principal Bimodules of Jacobson Radical of CSL Algebras

Ｓｔｒｏｎｇ－ｐｒｉｎｃｉｐａｌＢｉｍｏｄｕｌｅｓｏｆＪａｃｏｂｓｏｎＲａｄｉｃａｌ　ｏｆＣＳＬＡｌｇｅｂｒａｓＺｈｕＪｕｎ（朱军）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕｂｅｉＩｎｓｔｉｔｕｔｅｆｏｒＭａｔｉｏｎａｌｉｔｉｅｓ，Ｅｎｓ...     

格蕴涵代数中的扩张滤子

在格蕴涵代数中提出了扩张滤子的概念,讨论了扩张滤子与滤子,扩张滤子与素滤子,扩张滤子与滤子的根,扩张滤子与准素滤子,扩张滤子与最大滤子之间的关系.得到了扩张滤子的一些性质.最后,证明了在格H蕴涵代数中,扩张滤子与扩张滤子的根相等.     

Nest代数的Jacobson根的μ-核值保持映射

设算子代数Ａ　Ｂ（Ｈ），μ（Ａ）表示Ａ中的部分等距算子全体，若ｐ是Ａ到Ｂ（Ｈ）的线性映射，且对任意的ＵＥｕ（Ａ），有叫Ｕ）（ｋｅｃＵ）Ｇｒ“ｈＣｒ，则称　是Ａ上的μ－核值保持映射。本文将证明：Ｎｅｓｔ代数的Ｊａｃｏｂｓｏｎ根上的范数拓扑连续的μ－核值保持映射是广义内导子。     

A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x,y in H, there exists a natural number n = n(x,y) so that xⁿ and yⁿ have an LCM, that is, xⁿH \cap yⁿH is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H, or equivalently, H is weakly Krull, distinct elements of Q are v-coprime in H, or the radical of each element of Q is a maximal t-ideal of H. Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains.     

Short Kloosterman sums to powerful modulus

An estimate of Kloosterman sums with very few terms is obtained. The modulus corresponding to this sum is a powerful number, i.e., the radical (the product of all prime factors) of the modulus is very small in comparison with the modulus itself on the logarithmic scale. The proof essentially uses methods of A.G. Postnikov and H. Iwaniec.     

Invariant properties of representations under cleft extensions

The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.     

Plasma environment during hot cathode direct current discharge plasma chemical vapor deposition of diamond films

The plasma characteristics have been investigatedin situ by using optical emission spectroscopy (OES) and the Langmuir probe during hot cathode direct current discharge plasma chemical vapor deposition of diamond films. The changes of atomic H and CH radical in the ground state have been calculated quantitatively according to the results of OES and the Langmuir probe measurement as discharge current density varied. It is shown that atomic H and CH radicals both in the ground state and in the excited state increase with the enhancement of the discharge current density in the plasma. The electron density and CH emission intensity increase linearly with the enhancement of discharge current densities. The generation of different carbon-containing radicals is related to the elevation of electron temperature. Combining the growth process of diamond films and the diagnostic results, it is shown that atomic H in the excited state may improve the diamond growth efficiently, and the increase of electron temperature and density plays an important role in the increase of the deposition rate of diamond films. Project supported by the National Natural Science Foundation of China (Grant Nos. 69493505, 19475039, 19835002).     

Semiplane lattices over irreducible groups

Among transitive G-lattices we can distinguish a rather broad class of so-called semiplane lattices associated with the semidirect product of a Lie group H and a certain automorphism group G of it. It turns out that semiplane lattices are almost always plane in the irreducible case, i.e., we can take it that group H is commutative. An exception is the case of the adjoined representation of a simple Lie group. We have also proved that if group G is involutive and has a small radical, then all transitive G-lattices turn out to be semiplane.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 833–842, June, 1976.     

Locally perfect commutative rings are those whose modules have maximal submodules

R denotes a commutative ring. After Bass[B], a ring R is perfect in case every module has a projective cover. A ring R is a max ring provided that every nonzero i2-module has a maximal submodule. Bass characterized perfect rings as semilocal rings with T-nilpotent Jacobson radical J, and showed they are max rings. Moreover, Bass proved that R is perfect iff R satisfies the dec on principal ideals. Using Bass' theorems, the Hamsher-Koifman ([H],[K]) characterization of max R (see (3) ?(4) below), and the characterization of max R by the author via subdirectly irreducible quasi-injective R-modules, we obtain.     

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…     

Radicalizers

For a Kurosh–Amitsur radical class of rings, we investigate the existence, for a radical subring S of a ring A, of a largest subring T of A for which S is the radical. When T exists, it is called the radicalizer of S. There are no radical classes of associative rings for which every radical subring of every ring has a radicalizer. If a subring is the radical of its idealizer, then the idealizer is a radicalizer. We examine radical classes for which each radical subring is contained in one which is the radical of its own idealizer.     

On graded Thierrin radicals of graded rings

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring.     

分次反单本原根

本文给出分次情形的反单本原根SJGG,当群G满足|G|<∞,证明了SJG=SJref=SJG,其中SJref和SJG分别表示反射反单本原根和限制反单本原根.而且讨论了SJG的分次补根.     

群的遗传根性和强半单根性

本文利用群的根性的性质,解决了Szasz在环的根性理论中提出的公开问题在群论中的对应问题.同时我们研究了群的遗传根性和强半单根性的一些性质,并介绍了群的根类的交运算和并运算,由此得到了一些很好的结果.     

半遗传根的一个特征性质

本文证明了一个根性质成为半遗传的必要充分条件是它的亚直既约半单类是半规范的.证明了半遗传根的交根也是半遗传的.对亚直既约半单类为K的最小遗传根和半遗传根,分别给出了它们的低根构造.     

分次广义Γ-环的Brown-McCoy根

研究了分次广义Γ-环的弱强分次B row n-M cC oy根与拟弱强分次B row n-M cC oy根,并从不同角度刻划了分次广义Γ-环的弱强分次B row n-M cC oy根.证明了任何一个分次广义Γ-环都有弱强分次B row n-M cC oy根和拟弱强分次B row n-M cC oy根,而且弱强分次B row n-M cC oy根小于等于拟弱强分次B row n-M c-C oy根.     

J-半单亚直既约环类确定的上限

Ｆ．Ａ．Ｓｚａｓｚ在［１］中提出公开问题５５：设Ｋ是Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类,研究类Ｋ确定的上根．本文对此进行了研究,证明了Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类Ｋ确定的上根Ｒ是特殊根,它介于Ｊａｃｏｂｓｏｎ根与Ｂｒｏｗｎ－ＭｃＣｏｙ根之间．并给出任意结合环Ａ为Ｒ－根环的充要条件．