*(English)*Zbl 0591.16005

Let H be a finite dimensional Hopf algebra over the field k and let A be an H-module algebra. Then the smash product A#H is a ring extension of A and the ring ${A}^{H}$ of H-invariants defined by ${A}^{H}=\{a\in A|$ $h\xb7a=\u03f5\left(h\right)a\}$ is a subring of A. This paper studies the connection between these rings using the key fact that if H is semisimple, then it has a left integral, namely an element x with $hx=\u03f5\left(h\right)x$ for all $h\in H$, satisfying $\u03f5$ (x)$\ne 0\xb7$

Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if $W\subseteq V$ are A#H-modules, then W is an A-direct summand of V if and only if it is an A#H-direct summand.

Section 2 introduces a Morita context $[B{,}_{B}{A}_{A\#H}{,}_{A\#H}{A}_{B},A\#H]$ where $B={A}^{H}$ and with [, ] nondegenerate. Then criteria are given for (, ) to be nondegenerate and the context is used to relate ${A}^{H}$ to A#H. The paper also proposes an interesting question. Namely if H is a finite dimensional semisimple Hopf algebra and if A is semiprime, must A#H also be semiprime? Some partial results are given.

##### MSC:

16W30 | Hopf algebras (assoc. rings and algebras) (MSC2000) |

16D70 | Structure and classification of associative ring and algebras |