Invariant properties of modules under smash products from finite dimensional algebras

We give the relationship between indecomposable modules over the finite dimensional $k$-algebra $A$ and the smash product $A\mathrm{♯}G$ respectively, where $G$ is a finite abelian group satisfying $G\subseteq Aut(A)$, and $k$ is an algebraically closed field with the characteristic not dividing the order of $G$. More precisely, we construct all indecomposable $A\mathrm{♯}G$-modules from indecomposable $A$-modules and prove that an $A\mathrm{♯}G$-module is indecomposable if and only if it is an indecomposable $G$-stable module over $A$. Besides, we give the relationship between simple, projective and injective modules in $modA$ and those in $modA\mathrm{♯}G$.