  We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$,
$D \in {\mathcal C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are
semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra,
$M$, $N$ two $H$-(co)modules such that $M \otimes N$ is (co)semisimple as a
$H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective
$k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$)
is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of
all finite dimensional modules, comodules or Yetter-Drinfel'd modules over $H$
the dimension of which is invertible in $k$ are Serre categories.
