Use of "field or C.D.V.R." hypothesis in a theorem

Question

In Peter Webb's excellent book "A course in finite group representation theory", Theorem 11.6.1 states

Let $R$ be a field or a complete discrete valutation ring, and let $U$ be a indecomposable $RG$-module. Then there is a unique conjugacy class of subgroups $Q$ of $G$ that are minimal subject to the property that $U$ is relatively $Q$-projective.

Then he gives two proofs of this result, and both of them use the Mackey formula.

In the second proof, the hypothesis "$R$ has to be a field or a C.D.V.R" is used (it seems) in saying that the endomorphism ring of $U$ is local. However, in the first proof I really don't understand where it is used (i.e., what implication would fail if $R$ were just a commutative ring with $1$).

I would be willing to post the proof here but I am not sure it is okay to do it, can someone give me advice about it? There is, however, a pre-publication version freely available online on the author webpage

Answer 1

The first proof shows that $U$ is a summand of a direct sum of modules induced from subgroups of the form $K\cap^gH$. Up to there, the proof works for any $R$.

But the next step is to deduce that $U$ is a summand of one of these induced modules. That uses the Krull-Schmidt theorem, which is not true for general $R$.