Weakly associated prime filtration (Q700374)

Let \(A\) be a commutative ring, \(M\) an \(A\)-module and \(\text{Ass}_R(M)\) the set of (weakly) associated primes of \(M\) (recall that, \(P\in \text{Ass}_R(M)\) if \(P\) is a minimal prime over the annihilator of some \(x\in M\)). For an ideal \(I\) of \(A\), consider \(G(A)\) (respectively \(G(M)\)), the associated graded ring of \(A\) (respectively the associated graded module of \(M\)) with respect to \(I\). In the paper under review, the author proves that \(\text{Ass}_R(M)\) is finite if and only if \(M\) has a filtration \[ 0=M_0\subset M_1\subset \dots \subset M_n=M \] with \(\text{Ass}_R(M_i/M_{i-1})=\{P_i\}\), \(1\leq i\leq n\), and \(\text{Ass}_R(M)=\{ P_1,\dots,P_n\}\). He also proves that \(\bigcup_n \text{Ass}_R(M/I^{n+1}M)\) is finite, provided so is \(\text{Ass}_{G(A)}(G(M))\).