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.



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


