POSNER AND HERSTEIN THEOREMS FOR DERIVATIONS OF 3-PRIME NEAR-RINGS

The analog of Posner's theorem on the composition of two derivations i n prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a is an element of N is such that ax(d) = x(d)a for all x is a n element of N, then a is a central element. As a consequence it is sh own that if d(1) and d(2) are nonzero derivations of a 2-torsionfree 3 -prime near-ring N and x(d1)y(d2) = y(d2)x(d1) for all x,y is an eleme nt of N, then N is a commutative ring. Thus two theorems of Herstein a re generalized.