On the Index of Nilpotency of Semigroup Graded Rings

We find the index of nilpotency of a strong supplementary semilattice sum of rings, R=\tdsp\sum α∈ Y R α , where Y is a semilattice, when each R α has index of nilpotency≤ k . Then we find the index of nilpotency of R when it is graded over a rectangular band Y and each R α has index of nilpotency≤ 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