NOTES ON THE COMMUTATIVITY OF PRIME NEAR-RINGS

Abstract

Let N be a 3-prime right near-ring and let f be a generalized (theta, theta) - derivation on N with associated. (theta, theta) - derivation d: It is proved that N must be a commutative ring if d not equal 0 and one of the following conditions is satisfied for all x; y 2 N W. i / f. O x; y _ / D 0I. i i / f([x, y]) = theta([x, y]); (iii)f (x circle y) = 0 (i) f(xoy) = theta(x circle y); (v) f([x, y]) = theta(xoy); (vi) f(xoy) = theta([x, y]). We also prove theorems which assert that N is commutative, but not necessarily a ring.