Semigroups and rings whose zero products commute

Let S be a semigroup with zero 0 and let n greater than or equal to 2. We say that S satisfies ZC(n) if a(1) ... a(n) = 0 double right arrow a(sigma(1)) ... a(sigma(n)) = 0 for each permutation sigma is an element of S-n. A ring R satisfies ZC(n) if (R, .) satisfies ZC(n). We show that if S satisfies ZC(n) for a fixed n greater than or equal to 3, then S also satisfies ZC(n+1), but we give an example of a ring R with identity which satisfies ZC(2) but does not satisfy ZC(3). We show that a semigroup with no nonzero nilpotents satisfies ZC(n) for all n greater than or equal to 2 and investigate rings that satisfy ZC(n).