  We prove that every {finitely generated residually finite}-by-sofic group
satisfies Kaplansky's direct and stable finiteness conjectures with respect to
all noetherian rings. We use this result to provide countably many new examples
of finitely presented non-LEA groups, for which soficity is still undecided,
satisfying these two conjectures. Deligne's famous example of a non residually
finite group is among our examples, along with the families of amalgamated free
products SL_n(Z[1/p])*_{F_r}SL_n(Z[1/p]) and HNN extensions
SL_n(Z[1/p])*_{F_r}, where p>2 is a prime, n>2 and F_r is a free group of rank
r, for all r>1.
