Stable finiteness of group rings in arbitrary characteristic (Q1851239)

A ring \(R\) is said to be directly finite (resp. stably finite) if one-sided inverses in \(R\) (resp. in all matrix rings \(M_n(R)\)) are two sided. It has long been known that the group algebra \(K[G]\) of any group \(G\) over a field \(K\) of characteristic zero is directly finite [\textit{I. Kaplansky}, Fields and Rings, Chicago Lectures in Mathematics, Univ. of Chicago Press, Chicago (1970; Zbl 0238.16001); \textit{M. S. Montgomery}, Bull. Am. Math. Soc. 75, 539-540 (1969; Zbl 0174.31204); \textit{D. S. Passman}, Proc. Am. Math. Soc. 28, 371-374 (1971; Zbl 0217.07802)]. On the other hand the direct finiteness of \(K[G]\) in the case when the characteristic of \(K\) is \(p>0\) has been an open problem. In the present paper the authors prove that over a division ring \(D\) of any characteristic, and for any free-by-amenable group \(G\), the group ring \(D[G]\) is stably finite. This is achieved by showing that for each finitely generated subgroup \(S\) of such a group \(G\), the group ring \(D[S]\) admits a faithful Sylvester rank function which implies that \(D[G]\) is stably finite. The proof makes an interesting use of the translation rings associated with amenable groups.