10.40 (1986)
Open(H. Bass). Let $G$ be a group, $e$ an idempotent matrix over $\mathbb{Z}G$ and let $\operatorname{tr} e = \sum_{g \in G} e_g g$, $e_g \in \mathbb{Z}$.
$\qquad$ Strong conjecture: for any non-trivial $x \in G$, the equation $\sum_{g \sim x} e_g = 0$ holds where $\sim$ denotes conjugacy in $G$.
$\qquad$ Weak conjecture: $\sum_{g \in G} e_g = 0$.
Progress
The strong conjecture has been proved for finite, abelian and linear groups and the weak conjecture has been proved for residually finite groups.
H. Bass’ Comment of 2005: Significant progress has been made; a good up-to-date account is in the paper (A. J. Berrick, I. Chatterji, G. Mislin, Math. Ann., 329 (2004), 597–621).
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