• (σ - τ) - DERIVATIONS OF NEAR-FIELD SPACES OVER A NEAR-FIELD

Abstract

Let N be a left-near-field space and let s, t be automorphisms of N.  An additive mapping d: N ® N is called a ( s, t ) – derivation on N if d(xy) = s(x) d(y) + d(x)t(y) for all x, y Î N. In this paper, Dr N V Nagendram as author obtain Leibnitz’ formula for (s, t) – derivations on near-field spaces over a near-field which facilitates the proof of the following result. Let n ³ 1 be an integer, N be a n-torsion free and d a (s, t ) – derivation on N with dⁿ (N) = {0}. If both s, t commute with dⁿ for all n ³ 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(xsy) = 0, for all x, y Î N a near-field space over a near-field.