We show that every finitely
generated nilalgebra having nilalgebras of matrices is a homomorphic image of
nilalgebras constructed by the Golod method (Golod, 1965 and 1969). By applying
some elements of module theory to these results, we construct over any field
non-residually finite nilalgebras and Golod groups with non-residually finite
quotients. This solves Sunkov’s problem (Kourovka Notebook, 1995, Problem
12.102). Also, we reduce Kaplansky’s problem on the existence of a f.g. infinite
p-group G such that the augmentation ideal ωK[G] over a nondenumerable field K is
a nilideal (Kaplansky, 1957, Problem 9) to the study of the just-infinite quotients of
Golod groups.
