Finitely-generated modules over a Frobenius algebra

Question

Let $A$ be a Frobenius algebra over a field $k$ and let $P_1,\ldots, P_t$ be the principal indecomposable modules of $A_A$. If $W$ is a finitely-generated right $A$-module, then there exists $r>0$ such that $W^r\cong F\oplus E$ as $A$-modules, where $F$ is a free $A$-module is $E$ is a non-faithful $A$-module

This is actually the second part of a two-part lemma from Montgomery's Hopf Algebras and Their Actions on Rings (Lemma 3.1.1); I believe I've proved the first part. I mentioned the principal indecomposables because I think we need them, but I can't for the life of me figure this out.

Any help is appreciated. (I included the Hopf algebras tag because of the text this came from.)

Answer 1

It's a consequence of Part $1$ of the lemma which says:

$W$ is a faithful $A$-module if and only if each $P_i$ appears as a summand in the direct sum decomposition of $W$.

So look at the direct sum decomposition of $W$ and split off as many copies of $A$ as you can. Those copies form $F$, what's left over forms $E$. The fact that you couldn't split off another copy of $A$ means some $P_i$ must be missing, so use Part $1$ of the theorem on $E$.