Abstract

Let R be a prime ring with characteristics not 2 and <TEX>${\sigma},\;{\tau},\;{\alpha},\;{\beta}$</TEX> be auto-morphisms of R. Suppose that <TEX>$d_1$</TEX> is a (<TEX>${\sigma},\;{\tau}$</TEX>)-derivation and <TEX>$d_2$</TEX> is a (<TEX>${\alpha},\;{\beta}$</TEX>)-derivation on R such that <TEX>$d_{2}{\alpha}\;=\;{\alpha}d_2,\;d_2{\beta}\;=\;{\beta}d_2$</TEX>. In this note it is shown that; (1) If <TEX>$d_1d_2$</TEX>(R) = 0 then <TEX>$d_1$</TEX> = 0 or <TEX>$d_2$</TEX> = 0. (2) If [<TEX>$d_1(R),d_2(R)$</TEX>] = 0 then R is commutative. (3) If(<TEX>$d_1(R),d_2(R)$</TEX>) = 0 then R is commutative. (4) If <TEX>$[d_1(R),d_2(R)]_{\sigma,\tau}$</TEX> = 0 then R is commutative.