‎let $r$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$u$ the utumi quotient ring of $r$‎, ‎$c$ $(=z(u))$ the extended centroid‎ ‎of $r$‎. ‎let $0neq ain r$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $c$ which is noncentral valued on $r$‎. ‎suppose‎ ‎that $g$ and $h$ are two nonzero generalized derivations of $r$‎ ‎such that $a(h(f(x))f(x)-f(x)g(f(x)))in ...

In his classical paper 1 , Chang invented the notion of MV-algebra in order to provide an algebraic proof of the completeness theorem of infinite valued Lukasiewicz propositional calculus. Recently, the algebraic theory of MV-algebras is intensively studied, see 2–5 . The notion of derivation, introduced from the analytic theory, is helpful to the research of structure and property in algebraic...

Let R be a commutative ring with identity. By a Bres̃ar generalized derivation of R we mean an additive map g : R→ R such that g (xy) = g (x) y + xd (y) for all x, y ∈ R, where d is a derivation of R. And an additive mapping f : R → R is called a generalized derivation in the sense of Nakajima if it satisfies f(xy) = f(x)y + xf(y) − xf(1)y for all x, y ∈ R. In this paper we extend some results o...

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsio...

We characterize the principal differential ideals of a polynomial ring in n indeterminates with coefficients in the ring of differential polynomials in n indeterminates and derivation given by a “general” element of Lie(GLn) and use this characterization to construct a generic Picard-Vessiot extension for GLn. In the case when the differential base field has finite transcendence degree over its...

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with (N,+,×) if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in Z. Notably, by constructing a nontrivial...

Let $mathcal{A}$ be a $C^*$-algebra and $Z(mathcal{A})$ the‎ ‎center of $mathcal{A}$‎. ‎A sequence ${L_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{A}$ with $L_{0}=I$‎, ‎where $I$ is the‎ ‎identity mapping‎ ‎on $mathcal{A}$‎, ‎is called a Lie higher derivation if‎ ‎$L_{n}[x,y]=sum_{i+j=n} [L_{i}x,L_{j}y]$ for all $x,y in  ‎mathcal{A}$ and all $ngeqslant0$‎. ‎We show that‎ ‎${L_{n}}_{n...

‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0neq ain R$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))in ...