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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.

Quotients of nilalgebras and their associated groups