  We consider a family of finitely presented groups, called Universal Left
Invertible Element (or ULIE) groups, that are universal for existence of
one--sided invertible elements in a group ring K[G], where K is a field or a
division ring. We show that for testing Kaplansky's Direct Finiteness
Conjecture, it suffices to test it on ULIE groups, and we show that there is an
infinite family of non-amenable ULIE groups. We consider the Invertibles
Conjecture and we show that it is equivalent to a question about ULIE groups.
We also show that for any group G, direct finiteness of K[ G x H ] for all
finite groups H implies stable finiteness of K[G]. Thus, truth of the Direct
Finiteness Conjecture implies stable finiteness. By calculating all the ULIE
groups over the field K=F_2 of two elements, for ranks (3,n), n<=11 and (5,5),
we show that the Direct Finiteness Conjecture and the Invertibles Conjecture
(which implies the Zero Divisors Conjecture) hold for these ranks over F_2.
