Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good resources for this problem that gives some historical overview are:
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Passman, Donald S. The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.
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Passman, Donald S. Group rings, crossed products and Galois theory. CBMS Regional Conference Series in Mathematics, 64. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
The conjecture has been proven affirmative, when $G$ belongs to special classes of groups. I tried to write down some of the history:
- Ordered groups (A.I. Malcev 1948 and B.H. Neumann 1949)
- Supersolvable groups (E. Formanek 1973)
- Polycyclic-by-finite groups (K.A. Brown 1976, D.R. Farkas & R.L. Snider 1976)
- Unique product groups (J.M. Cohen, 1974)
Here are my questions:
- Was Irving Kaplansky the first one to state this conjecture? Can someone provide me with a reference to a paper or book that claims this?
- Since the publications of Passman's expository note (above) in 1986, has there been any major developments on the problem? Are there any new classes of groups that will yield a positive answer to the conjecture? Can someone help me to extend my list above?
The zero-divisor conjecture (let's denote it by "(Z)") is related to the following two conjectures:
(I): If $G$ is torsion-free, then $K[G]$ has no non-trivial idempotents.
(U): If $G$ is torsion-free, then $K[G]$ has no non-trivial units.
Now, if $G$ is torsion-free, then one can show that:
(U) $\Rightarrow$ (Z) $\Rightarrow$ (I).
Has there been any developments, since 1986, to any partial answers on conjecture (U)? Passman claims that "this is not even known for supersolvable groups". Is this still the case?
I want to point out that this post is related to another old MO-post.
Best Answer
I suggest you have a look at Chapter 10 of W. Lueck's book, "$L^2$-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002. There he discusses the Atiyah conjecture (conjecture 10.3): if $K$ is a subfield of $\mathbb{C}$, a group $G$ satisfies the Atiyah conjecture with coefficients in $K$ if for any $m\times n$ matrix $A$ with coefficients in $K[G]$, the von Neumann dimension of the kernel of the operator $r_A:\ell^2(G)^m\rightarrow \ell^2(G)^n:x\mapsto Ax$, is an integer.
Lueck then proves (lemma 10.15) that if $G$ is torsion-free and satisfies the Atiyah conjecture with coefficients in $K$, then it satisfies Kaplansky's conjecture (Z). For $K=\mathbb{C}$ and $G$ amenable, the converse is true (lemma 10.16).
Lueck goes on to prove (Theorem 10.19) a remarkable result by P. A. Linnell (Division rings and group von Neumann algebras. Forum Math., 5(6):561-576,1993): Let $\cal{C}$ be the smallest class of groups containing free groups and closed under directed unions and extensions with elementary amenable quotients; if $G$ is in $\cal{C}$ and has finite subgroups of bounded order, then the Atiyah conjecture with coefficients in $\mathbb{C}$ holds.
For further reading and more recent results, try typing "Atiyah conjecture" on Google or in the ArXiV.