$0 :_M I^n$ is finitely generated for all $i\ge 1$?

Question

I see the remark that: "Let $R$ be a Noetherian commutative ring, $M$ an $R$-module and $I$ an ideal of $R.$ Assume that $0 :_M I$ is finitely generated. Then $0 :_M I^n$ is finitely generated for all $i\ge 1.$" Could someone help me some ideas to prove this? Thank you very much.

Answer 1

Let $I=(x_1,..,x_k)$. Then for $n \geq 1, (0 :_M I^{n+1})/(0 :_M I) \hookrightarrow \oplus x_i(0 :_M I^{n+1})\hookrightarrow \oplus(0 :_M I^{n})$, so by induction on $n, (0 :_M I^{n+1})/(0 :_M I)$ and thus also $(0 :_M I^{n+1})$ is finitely generated.