Document Type : Original Article
School of Mathematics and Computer Science, Damghan University, P. O. Box 36716, Damghan 41167, Iran
Let $A$ and $X$ be Banach algebras such that $X$ is a Banach algebraic $A$-module. The generalized module extension $A\bowtie X$, which is a strongly splitting Banach algebra extension of $X$ by $A$, was recently introduced and studied. Many known Banach algebras such as module extension, Lau product, (generalized) semidirect product and also the direct product of Banach algebras have this general framework. We first investigate the $n$-weak amenability of $A\bowtie X$ and then we characterize its cyclic amenability improving some known results. We also examine our results to some special generalized module extension algebras. Furthermore, we characterize the cyclic amenability of some concrete Banach algebras related to locally compact groups.