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:
Passman, Donald S. The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.
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 (?)
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.