Suppose that σ is homomorphism on Banach algebra A. Then in this paper we introduce and study the new two notions σ-cyclic derivation and σ-cyclic amenability for A. We investigate the relation between trace extension property and σ-cyclic amenability; indeed we show that the σ-cyclic amenability of  AI    implies that I has the trace extension property. Next, prove that the it’s converse can be true under the special conditions. One of the important result is that every σ-cyclic amenable is essential. Furthermore, for enery closed two-sided ideal I of A, the relation between of σ-cyclic amenability of A and σ   -cyclic amenability of  AI    has been studied. Also, we show that the σ-cyclic amenability of  A and AI    is equivalent. Finally, we study this notion on θ-Lau algebras and We investigate its relation with the similar concept on algebras A and B.