We assume throughout that all rings are commutative with 1 ?= 0. Prime ideals play a central role in commutative ring theory. Suppose that R is a ring. We recall that a prime ideal P of R is a proper ideal with the property that for a,b ? R,ab ? P implies a ? P or b ? P. A weakly prime ideal is a proper ideal P of R with the property that for a,b ? R,0 ?= ab ? P implies a ? P or b ? P. So a prime ideal is weakly prime. But a weakly prime ideal need not be prime; and by definition 0 is always weakly prime. In this thesis, we consider some generalizations of these notions: Suppose that n ? 2 is a positive integer. A proper ideal P of R is called (n?1,n)-prime ideal if for a?,a?,...,a n ? R ? R, a?a?...a n ? P implies a?a?...a i-1 a i+1 ...a n ? P for some i ? {1,2,...,n}. Also a proper ideal P of R is called (n ? 1,n)-weakly prime if for a?,a?,...,a n ? R,0 ?a?a?...a n ? P ? P implies a?a?...a i-1 a i+1 ...a n ? P for some i ? {1,2,...,n}.