Lie triple derivations on primitive rings

In this paper, we show that for each Lie triple derivation L on primitive ring R of characteristic not 2 with nontrivial idempotent, there exists an ordinary derivation D of R into a primitive ring [Formula: see text] containing R and additive mapping λ of R into the center of [Formula: see text] that annihilates commutators such that L(X) = D(X) + λ(X).

ON SUPERNILPOTENT RADICALS WITH THE AMITSUR PROPERTY

A radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

The Nanostructures of Amorphous Silicas

Topologically modeled amorphized silica structures have been refined using a molecular dynamics simulation technique. Several metastable structures with substantially different medium-range connectivities, as characterized by primitive ring statistics, were obtained. Whereas the total correlation function is insensitive to these differences, the first sharp diffraction peak derived from energy-filtered electron diffraction shows a promising correlation to medium-range structure.

On simple, primitive and prime rings relative to a torsion theory

For a hereditary torsion theory τ, we show by examples that the concepts of τ-Artinian τ-simple, τ-Artinian τ-primitive, τ-Artinian τ-prime, τ-Artinian τ-semiprimitive, and τ-Artinian τ-semiprime rings are different from each other and thus answer a question raised by Bland in his book [Topics in Torsion Theory, (Mathematical Research, 103, Wiley-VCH, 1998]. The example of a τ-Artinian τ-primitive ring which is not τ-simple here appears to be a counter-example to a result of Bland in the same publication.

Homogeneous Polynomials, Centralizers and Derivations in Rings

Letdbe a non-zero derivation on a primitive ringRandƒ(x1,…,xn) a homogeneous polynomial of degreem. We prove that the conditiond(ƒ(r1,…,rn)t) = 0, for allr1,…,rn∈R, withtdepending onr1,…,rn, forcesRto be a finite dimensional central simple algebra andƒpower-central valued onR. We also obtain bounds on [R:Z(R)] in terms ofm.