On (?, ?) derivations with module values [Modül de?erli? (? ?)-türevler üzeri?ne]

Abstract

Let R be a ring, X ? (0) an R-bi-module, d : R ? Xa(a,r)- derivation with module value such that d? = ?d, d? = ?d and U ? (0) an ideal of R. Furthermore the following properties are also satisfied. For x ? X, a ? R xRa = 0 implies x = 0 or a = 0 . . . . . . (G1) For a ? R, x ? X aRx = 0 implies a = 0 or x = 0 . . . . . . (G2) In this paper we have proved the following results; (1) If (G1) (or (G2)) is satisfied and for a ? R, d(U)a = 0 (or ad(U) = 0) then d = 0 or a = 0 (2) If (G1) is satisfied and [X, U] ? C(X) or [X, U]?,? ? C?,?(X) then R is commutative (3) Let X be a2-torsion free R-bi module, d1 : R ? Xa(?, ?)-derivation, d2 : R ? R a derivation such that d2 (U) ? U. If (G1) is satisfied and d1d2(U) = 0 then d1 = 0 or d2 = 0 (4) Let X be a2-torsion free R-bi-module. If (G1) and (G2) are satisfied and for a ? U, [d(U),a] ?,? ? C?,?(X) then a ? Z or d = 0. © TÜBİTAK.