On the index of nilpotency of semigroup graded rings (Q5930169)

If \(R=\sum\{R_\alpha\mid\alpha\in Y\}\) is a strong supplementary semilattice sum of rings over a semilattice \(Y\), then \(R\) has index of nilpotency \(\leq k\) if and only if each \(R_\alpha\) has index of nilpotency \(\leq k\). For a ring \(R\) with bounded index of nilpotency \(k\), we have the following bounds on the index of nilpotency of a semigroup ring \(R[S]\): (1) \(k+1\) if \(S\) is a right zero semigroup, and (2) \(k+2\) when \(S\) is a rectangular band. If \(R=\sum\{R_e\mid e\in E\}\) is a strong band graded ring over a normal band \(E\), and if each \(R_e\) has index of nilpotency \(\leq k\), then \(R\) has index of nilpotency \(\leq k+2\). Examples are given to show that these are the best possible bounds, and some special generalizations of these results are given.