Generalized Derivations on Prime Near Rings
Let be a near ring. An additive mapping is said to be a
right generalized (resp., left generalized) derivation with associated derivation on if (resp., ) for all . A mapping
 is said to be a generalized derivation with associated derivation on if
 is both a right generalized and a left generalized derivation with associated derivation
 on . The purpose of the present paper is to prove some theorems in the setting of
a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby
extending some known results on derivations.
Generalized derivations on prime near rings II
Let $$N$$N be a near ring. An additive mapping $$f: N\longrightarrow N$$f:N⟶N is said to be a right generalized (resp. left generalized) derivation with associated derivation $$d$$d on $$N$$N if $$f(xy)=f(x)y+xd(y)$$f(xy)=f(x)y+xd(y) (resp. $$f(xy)=d(x)y+xf(y)$$f(xy)=d(x)y+xf(y)) for all $$x,y\in N.$$x,y∈N. A mapping $$f:N\longrightarrow N$$f:N⟶N is said to be a generalized derivation with associated derivation $$d$$d on $$N$$N if $$f$$f is both a right generalized and a left generalized derivation with associated derivation $$d$$d on $$N$$N. The purpose of the present paper is to prove some theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a pair of suitably-constrained generalized derivations, thereby extending some known results on derivations. We show that if $$N$$N is 2-torsion free and $$f_1$$f1 and $$f_2$$f2 are generalized derivations such that $$f_1f_2=0$$f1f2=0, then $$f_1=0$$f1=0 or $$f_2=0$$f2=0; we prove other theorems asserting triviality of $$f_1$$f1 or $$f_2$$f2; and we also prove some commutativity theorems.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.