On the index of nilpotency of semigroup graded rings

We find the index of nilpotency of a strong supplementary semilattice sum of rings, R = Sigma (alpha is an element ofY) R-alpha, where Y is a semilattice, when each R-alpha has index of nilpotency less than or equal to k. Then we find the index of nilpotency of R when it is graded over a rectangular band Y and each R-alpha has index of nilpotency less than or equal to k. These results are generalized to normal band graded rings. Further, we find sufficient conditions for a ring graded by a semilattice of nilpotent semigroups to have bounded index of nilpotency. We also show by examples that these conditions are necessary in some cases.