Let A be a bialgebra and let S be a right A-comodule algebra which has an A-comodule subalgebra T with common identity. We show that if S is a separable extension of T, then for a left A-module algebra K, K♯S is a separable extension of K♯T. Similar result holds for left A-module algebras and right A-comodule algebras.
Separable extension
smash product
module algebra and comodule algebra