On Skew Derivations in Semiprime Rings

We investigate the commutativity in a (semi-)prime ring R which admits skew derivations δ ₁, δ ₂ satisfying [δ ₁(x), δ ₂(y)]?=?[x, y] for all x, y in a nonzero right ideal of R. This result is a natural generalization of Bell and Daif’s theorem on strong commutativity preserving derivations and a recent result by Ali and Huang.     

微商共同作用在半素环的Lie理想上的结果

本文讨论了微商共同作用在半素环的某个Ｌｉｅ理想上的问题。给出了如下结果：设Ｒ是带有中心Ｚ（Ｒ）的半素环,Ｑｍｒ是Ｒ的极大右商环,Ｌ是Ｒ的非交换Ｌｉｅ理想,ｄ和δ是Ｒ的微商,假设ｒＲ（「Ｌ,Ｌ」）＝０且ｄ（ｘ）ｘ－ｘδ（ｘ）∈Ｚ（Ｒ）对任意ｘ∈Ｌ成立,则在Ｒ的扩张形心Ｃ中存在一个幂等元ｅ使得ｄ（１－ｅ）Ｑｍｒ＝０和δ（１－ｅ）Ｑｍｒ）＝０并且ｅＱｍｒ满足Ｓ４。另外给出微商共同作用在半素环上多项式的结     

A Note on Derivations in Semiprime Rings

Throughout,R will be semiprime ring,Qmr=Qmr(R) the maximal right quotient ringof R,and C=Z(Qmr) the extended centroid of R(see[1 ,Chapter2 ] for details) .　 In [2 ] ,Bresar,M.discussed the identity of derivations in a prime ring,and provedthat if nonzero derivations d and g of R satisfy d(x) g(y) =g(x) d(y) for all x,y∈R,then there exists aλ in C such thatg(x) =λd(x) for all x∈ R.Itis natural to ask whatresults can be obtained if d(x) g(x) =g(x) d(x) for all x∈ R.In[3 ] ,Bresa…     

On the Derivations of Semiprime Rings and Noncommutative Banach Algebras

Abstract 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). The author has been supported by Kangnung National University, Research Fund, 1998     

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, RF be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of RE. Furthermore, if there exists a fixed positive integer n such that μ(x)^n = 0 for all x∈I, then μ=0.     

Semiprime Lie Rings of (Anti-)Symmetric Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with involution, and δ a nonzero derivation of R. Let S be the set of symmetric elements in R, and let K be the set of anti-symmetric elements in R. In this article, we investigate the semiprimeness of the Lie rings Sδ when δ is symmetric and Kδ when δ is anti-symmetric.     

A Note on Constants of Integral Derivations in Semiprime Rings

Let R be a semiprime ring with extended centroid C and with Q its Martindale symmetric ring of quotients. Suppose that δ: R → R is a C-integral derivation. For a subring A of Q, let A^(δ) denote the subring of constants of δ in A. We prove that R^(δ) and Q^(δ) satisfy the same polynomial identities with coefficients in C. In particular, R^(δ) is not nil of bounded index.     

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.     

On Semiprime Multiplication Modules over Pullback Rings

We classify all those indecomposable semiprime multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [9 Ebrahimi Atani , S. , Farzalipour , F. ( 2009 ). Weak multiplication modules over a pullback of Dedekind domains . Colloquium Math. 114 : 99 – 112 .[Crossref] , [Google Scholar]] to a more general semiprime multiplication modules case.     

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.