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Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$. The wiki page: http://en.wikipedia.org/wiki/Group_ring says "This conjecture (zero divisor conjecture) is equivalent to K[G] having no non-trivial idempotents under the same hypotheses for K and G. " Is this obvious true? Are there some reference for this claim? |
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Passman showed that whenever there are zero-divisors in a group ring one also has (non-zero) nilpotent elements. He shows that for any field $k$ and any torsionfree group $G$, the ring $kG$ is a prime ring, i.e. the zero-ideal is a prime ideal. Now, if $a,b \in kG$ are non-zero and $ab=0$, then there exists $c \in kG$ such that $bca\neq 0$. (This is just another way of saying that the product of the two-sided ideal $(b)$ with the two-sided ideal $(a)$ cannot give the zero ideal, since $(0)$ is a prime ideal.) Now, we see that $(bca)^2 = bcabca =0$. Hence, $kG$ contains nilpotent elements. The other direction is obvious. |
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Clearly one implies the other as $x^2=x$ means $x(x-1)=0$. I doubt they are known to be equivalent since the sources I found: the K-theory handbook and Alain Valette survey (see Conjecture 2) listed them as separate conjectures. Of course, it would be hard to prove they are not equivalent, since both might be true! |
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