Let R be a commutative ring with identity. A proper ideal P of R is an (n - 1, n)-Fm-Prime ((n - 1, n)-weakly Prime) ideal if a1, … , anÎR, a1 … anÎP\P m (a1 … anÎP\{0}) implies a1 … ai-1 ai+1 … anÎP, for some iÎ{1, …, n}; (m, n³2).In this paper several results concerning (n - 1, n)-Fm-Prime and (n - 1, n)-weakly Prime ideals are proved. We show that in a Noetherian domain a Fm-Prime ideal is primary and we show that in some well-known rings (n - 1, n)-Fm-Prime ideals and (n - 1, n)-Prime ideals coincide.