Generalized Derivations with Commutativity and Anti-commutativity Conditions

Let R be a prime ring with 1, with char(R) &#8800; 2; and let F : R &#8594; R be a generalized derivation. We determine when one of the following holds for all x,y &#8712; R: (i) [F(x); F(y)] = 0; (ii) F(x)&#927;F(y) = 0; (iii) F(x) &#927; F(y) = x &#927; y .</p

Higher Derivatives and Finiteness in Rings

&lt;p&gt;Let n be a positive integer, R a prime ring, U a nonzero right ideal, and d a derivation on R. Under appropriate additional&#12288;hypotheses, we prove that if d&lt;sup&gt;n&lt;/sup&gt;(U) is finite, then either R is finite or d is nilpotent. We also provide an extension to semiprime rings.&lt;/p&gt;</p

Some conditions for finiteness and commutativity of rings

We present several new sufficient conditions for a ring to be finite; we give two conditions which for periodic rings R imply that R must be either finite or commutative; and we study commutativity in rings with only finitely many non-central subrings

A subsystem-independent generalization of entanglement

We introduce a generalization of entanglement based on the idea that entanglement is relative to a distinguished subspace of observables rather than a distinguished subsystem decomposition. A pure quantum state is entangled relative to such a subspace if its expectations are a proper mixture of those of other states. Many information-theoretic aspects of entanglement can be extended to the general setting, suggesting new ways of measuring and classifying entanglement in multipartite systems. By going beyond the distinguishable-subsystem framework, generalized entanglement also provides novel tools for probing quantum correlations in interacting many-body systems.Comment: 5 pages, 1 encapsulated color figure, REVTeX4 styl

On Prime Near-Rings with Generalized Derivation

Let N be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center Z. We prove that if N admits a nonzero generalized derivation f such that f(N)⊆Z, then N is a commutative ring. We also discuss some related properties

On Generalized Periodic-Like Rings

Let R be a ring with center Z, Jacobson radical J, and set N of all nilpotent elements. Call R generalized periodic-like if for all x∈R∖(N∪J∪Z) there exist positive integers m, n of opposite parity for which xm−xn∈N∩Z. We identify some basic properties of such rings and prove some results on commutativity