Let k be a field of characteristic zero, n any positive integer and let δn be the derivation ∑n i=1Xi ∂ ∂Yi of the polynomial ring k[X1, . . . , Xn, Y1, . . . , Yn] in 2n variables over k. A Conjecture of Nowicki (Conjecture 6.9.10 in (8)) states the following ker δn = k[X1, . . . , Xn, XiYj −XjYi; 1 ≤ i < j ≤ n] in which case we say that δn is standard. In this paper, we use the elimination th...

Let k[X] be the algebra of polynomials in n variables over a field k of characteristic zero, and let f ∈ k[X] r k. We present a construction of a derivation d of k[X] whose ring of constants is equal to the integral closure of k[f ] in k[X]. A similar construction for fields of rational functions is also given.

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free ∗-prime ring with involution is a generalized left derivation.

Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of (α, β)-derivation to (α, β)higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for α and β is (α, β)-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then...

Copyright q 2012 Kalyan Kumar Dey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let N be a 2-torsion free prime Γ-near-ring with center ZN. Let f, d and g, h be two generalized derivations on N. We prove the following res...

A New Continuous Distribution and Two New Families of Distributions Based on the Exponential Recent work on social status led to derivation of a new continuous distribution based on the exponential. The new variate, termed the ring(2)-exponential, in turn leads to derivation of two closely-related new families of continuous distributions, which we call the mirrorexponential and the ring-exponen...

Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R, f(x 1,…, x n ) a noncentral multilinear polynomial over C in n noncommuting variables, and a, b ∈ R such that a[F(f(r 1,…, r n )), f(r 1,…, r n )]b = 0 for any r 1,…, r n ∈ R. Then one of the following holds: (1) a = 0; (2) b = 0; (3) ...