The number of homomorphic images of certain Golod groups and algebras (Q1921415)

An algebra \(A\) is called a Golod \(d\)-algebra (\(d>1\)), if it is \(d\)-generated and nonnilpotent, but any \((d-1)\)-generated subalgebra of \(A\) is nilpotent. Let \(F\) be a free algebra over a field \(k\), \(I\) be an ideal, generated by a sequence of polynomials \(f_1,f_2,\dots\), Golod proved that under the well known Golod condition the algebra \(A/I\) is infinite dimensional. Theorem 1. For any finite set \(M\) of integers greater or equal to \(d\), there exists a Golod \(d\)-algebra such that for any \(d'\) from \(M\), \(A\) contains a Golod \(d'\)-algebra (ideal) \(B\), with the Golod condition for \(A/B\). Theorem 2. For any \(d>1\) and any field there exists a Golod \(d\)-algebra having a continuum of residually finite pairwise nonisomorphic homomorphic images.