The Jacobson Radical of (R(G])~G

In this paper,the Jacobson radical of (R[G])^G, the fixed point subring of a group ring (R[G]) is studied. Some related properties of (R[G])^G are discussed.

TENSOR PRODUCTS OF JACOBSON RADICALS IN NEST ALGEBRAS

This paper studies the tensor product RN RM of Jacobson radicals in nest algebras, and obtains that RN RM = {T∈B(H1 H2) : T(N M)(?)N_ M_, N∈N,M∈M}; and based on the characterization of rank-one operators in RN RM,it is proved that if N, M are non-trivial then RN RM=R if and only if N, M are continuous.

The Jacobson and Brown-McCoy Radicals of Certain Group Rings

In this paper, we determine the Jacobson radicals and Brown-McCoy radicals of group rings of certain non-abelian groups and generalize some known results.

On_λ[S] Semigroup of Semigroup

In this paper,we introduce the notion of λ[S]([S] ρ) semigroup with the inner left and right translations of semigroup S,hence define the homomorphic mapping λ from λ[S] to S and we have λ[S]/kerλS λS and λ(J( λ[S])=J(S λ),J(S λ) is the Jacobson radical of S λ.

The Upper Radical Determined by the Class of all J-Semisimple Subdirectly Irreducible Rings

F.A.Szasz has put forward the open problem 55 in [1]: Let K be the class of all subdirectly irreducible rings, whose Jacobson radical is (0). Examine the upper radical determined by the class K. In this paper, the problem has been examined. (1) It has been proved that the upper radical R determined by the class K is a special radical,which lies between Jacobson radical and Brown-McCoy radical. (2) It has been given some necessary and sufficient condition of ring A to be an R-radical ring.

On Radicals of Skew Inverse Laurent Series Rings

Let R be a ring and a an automorphism of R. Amitsur proved that the Jacobson radical J（R[x]） of the polynomial ring R[x] is the polynomial ring over the nil ideal J（R[x]）∩R. Following Amitsur, it is shown that when R is an Armendaxiz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x; α], R[x,x^-1;α], R[[x^-1;α]] and R（（x^-1; α））, then （S） = （R）S = Nil（S）, （S） ∩ R = Nil（R）, where is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

Classical Radicals of Invariants of Algebraic Skew Derivations

Let σ be an automorphism and δ be a q-skew σ-derivation of an F-algebra A.We prove that if A is semiprimitive and δ is algebraic,then the subalgebra A^(δ)={r∈A|δ(r)=0}has nilpotent Jacobson radical.Using this result,we obtain similar relations for the Baer prime radical,the Levitzki locally nilpotent radical,and the Kothe nil radical when the field F is uncountable.Then we apply it to actions of the n^(2)-dimensional Taft Hopf algebra T_(n2)(q)and the q-analogue U_(q)(sl(2))of the enveloping algebra of the Lie algebra sl(2).

Pseudo-Injective Modules and Principally Pseudo-Injective Modules

Considered in this paper are pseudo-injective modules and principally pseudoinjective modules, which are generalizations of quasi-injective modules and PQ-injective modules. Pseudo-injective modules are dual to pseudo-projective modules. We study their properties and endomorphism rings, and obtain some properties of the Jacobson radical of such rings.

The Frattini Subalgebra of n-Lie Algebras

In this paper, we mainly study the structural notions of Frattini subalgebra and Jacobson radical of n-Lie algebras. The properties on Frattini subalgebra and the relationship between the Frattini subalgebra and Jacobson radical are studied. Moreover, we also study them when an n-Lie algebra is strong semi-simple, k-solvable and nilpotent.

Some Subsystems of a Lie Triple System Closely Related to Its Frattini Subsystem

The main purpose of the present paper is to give some properties of the Jacobson radical, the Frattini subsystem and c-ideals of a Lie triple system. Some further results concerning the Frattini subsystems of nilpotent and solvable Lie triple systems are obtained. Moreover, we develop inititally c-ideals for a Lie triple system and make use of them to give some characterizations of a solvable Lie triple system.

About Operator Weighted Shift

Abstract Let H be a complex seperable Hilbert space and ^（Jt^） denote the collection of bounded linear operators on H. For an operator in L（H）, rad{A}＇ denotes the Jacobson radical of the commutant of A. This paper characterizes the similarity of strongly irreducible operator weighted shift in terms of {A}＇/rad{A}＇. Moreover, we suggest some ways to determine when an operator weighted shift is strongly irreducible and when its commutant is commutative.

On the Derivations of Semiprime Rings and Noncommutative Banach Algebras

Let A be a noncommutative Banach algebra.Suppose there exists a continuous linear Jordan derivation D:A→A such that [D(x),x]D(x)[D(x),x]∈ rad(A) for all x ∈ A.In this case, D(A)rad(A).

Skew Incidence Rings and the Isomorphism Problem

Let X be a finite partially ordered set,R an associative unital ring andσan endomorphism of R.We describe some properties of the skew incidence ring I(X,R,σ)such as invertible elements,idempotents,the Jacobson radical and the center.Moreover,if two skew incidence rings I(X,R,σ)and I(Y,S,τ)are isomorphic and the only idempotents of R and S are the trivial ones,we show that the partially ordered sets X and Y are isomorphic.

Noncommutative Versions of the Singer-Wermer Conjecture with Linear Left θ-derivations

The noncommutative Singer-Wermer conjecture states that every linear （possibly unbounded） derivation on a （possibly noncommutative） Banach algebra maps into its Jacobson radical. This conjecture is still an open question for more than thirty years. In this paper we approach this question via linear left θ-derivations.

The Necessary and Sufficient Condition for Strong Irreducibility of Cowen-Douglas Operators

In this note, we show that a Cowen-Douglas operator is strongly irreducible if and only if its commutant algebra rood its Jocobson radical is isomorphic to a closed subalgebra of H^∞ （D）, where D is the open unit disk, and H^∞（D） denotes the collection of bounded holomorphic functions on D.