Mohammad Ashraf

Let $\mathcal{S}$ and $\mathcal{T}$ be two complex unital *-algebras. For any $H_1,H_2,\dots ,H_n\in \mathcal{S},$ define $g_1\left(H_1\right)=H_1,$$g_2(H_1,H_2)=H_1\diamond H_2=H_1{H}_2^{*}+H_2{H}_1^{*}$ and$$g_n(H_1,H_2,\dots ,H_n)=g_{n-1}(H_1,H_2,\dots ,H_{n-1})\diamond H_n$$

Let $\U$ be a von Neumann algebra with a projection $P\in \U$. For any $A_1,A_2,\ldots,A_n\in\U,$ define $p_1(A_1)=A_1,$ $p_n (A_1,A_2,\ldots,A_n)=[p_{n-1} (A_1,A_2,\ldots,A_{n-1}),A_n]$ for all integers $n\geq 2,$ where $[A,B]=AB-BA$ $(A,B\in\U)$ denotes the usual Lie product. Assume that $φ:\U\to\U$ is an additive mapping satisfying \[φ(p_n(A_1, A_2, \ldots, A_n)) = p_n(φ(A_1), A_2, \ldots, A_n) = p_n(A_1, φ(A_2), \ldots, A_n) \] for all $A_1, A_2, \ldots, A_n \in \U$ with $A_1A_2=P$ In this article, it is shown that the map $φ$ is of the form $φ(A)=WA+ξ(A)$ for all $A\in \U$, where $W\in \mathrm{Z}(\U)$, and $ξ:\U \to \Z(\U)$ ($\Z(\U)$ is the center of $\U$) is an additive map such that $ξ(p_n(A_1, A_2, \ldots, A_n) )=0$ for any $A_1, A_2, \ldots, A_n \in \U$ with $A_1A_2=P$. As an application, we characterize generalized Lie $n$-derivations on arbitrary von Neumann algebras.

Let $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring $${\mathfrak {R}}.$$ R . Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n , based on the deletion distance over the ring $${\mathfrak {R}}.$$ R . We also construct a bijection $$\Gamma$$ Γ between the elements of the ring $${\mathfrak {R}}$$ R and $$S_{D_{16}}.$$ S D 16 . As an application of $$\Gamma ,$$ Γ , the reversibility problem which occurs in DNA k -bases has been solved. Moreover, we introduce a Gray map $$\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}$$ Ψ hom : R n → F 2 8 n with respect to homogeneous weight $$w_{\hom }$$ w hom over the ring $${\mathfrak {R}}$$ R . Further, we discuss the GC -content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.

Let $N$ be a left near-ring and let $\sigma, \tau$ be automorphisms of $N$. An additive mapping $d : N \longrightarrow N$ is called a $(\sigma, \tau)$-derivation on $N$ if $d(xy) = \sigma (x)d(y) + d(x)\tau (y)$~for all $x,y \in N$. In this paper, we obtain Leibniz' formula for $(\sigma, \tau)$-derivations on near-rings which facilitates the proof of the following result: Let $n \geq 1$ be an integer, $N$ be $n$-torsion free, and $d$ a $(\sigma,\tau)$-derivation on $N$ with $d^{n}(N)=\{0\}$. If both $\sigma $ and $\tau$ commute with $d^{n}$ for all $n \geq 1$, then $d(Z)=\{0\}$. Further, besides proving some more related results, we investigate commutativity of $N$ satisfying either of the properties: $d([x,y])= 0,$ ~or~ $d(xoy)= 0$, for all $ x,y \in N$. 2000 Mathematics Subject Classification. 16W25, 16Y30