On examples of associative nil algebras (Q1907432)

The author proves the following main results: i) for any integer \(d\geq 2\), any finite set \(M\) of positive integers that contains, together with any number, all of its divisors, and an arbitrary associative commutative ring \(\Phi\) with 1, there exists a \(d\)-generated associative graded nil \(\Phi\)-algebra \(A\) such that 1) \(A\) is a nonnilpotent algebra; 2) for each \(m\in M\) (\(d^m-1\))-generated subalgebras of \(A^m\) are nilpotent; ii) for any integer \(d>2\) there exists a \(d\)-generated associative nonnilpotent nil algebra \(A\) such that \(A=I+J\) (\(I\triangleleft A\), \(J\triangleleft A\)) and \(d\)-generated subalgebras in \(I\) and \(J\) are nilpotent. Considering the associated group of the algebra, the author obtains the following corollary for groups: there exists a nonnilpotent \(d\)-generated group (\(d\geq 2\)) that is factorable into the product of two normal subgroups in which \(d\)-generated subgroups are nilpotent.