Weakly Associated Prime Filtration

Let R be a (not necessarily Noetherian) commutative ring and let M be a (not necessarily finitely generated) R-module. We characterize the modules with only finitely many weakly associated primes as those modules M admitting a chain 0 = M₀ M₁ ... M_n = M of submodules together with prime ideals p₁, p₂,...,p_n such that the set of weakly associated primes of M_i/M_i-1 is equal to {p_i} for all 1 i n. Let M = gr_a(M) = _n0aⁿM/aⁿ⁺¹M be the corresponding graded module over the graded ring R = gr_a(R) = _n0aⁿ/aⁿ⁺¹. It is shown that the union of the set of weakly associated primes of.....