Bounds for indices of nilpotency and nility
Let \(f(n,r)\) be the index of nilpotency of the generic \(r\)-generated nil ring of index \(n\). The author shows that \(f(n,r)<\tfrac 16n^6r^n\). Let \(g(n)\) be the index of nilpotency of the ideal generated by the \(m\)-th power of an \(m\)-set in the infinitely generated generic nil ring of index \(n\), where \(m=\lfloor\tfrac n2\rfloor\). Then the author obtains a bound \(f(n,r)<mg(n){r\choose m}\). The author shows further that \(f(n,r)<\tfrac 1{(m-1)!}n^{n^3}r^m\), a bound which is much smaller than Belov's bound if \(r\) is relatively large in comparison with \(n\). Several interesting consequences are also obtained.
scientific article; zbMATH DE number 1437111