Let $M$ be a finitely generated $R$-module, and $N$ be a submodule of $M$; is it possible to have a meaning for $Ann(M)/N$ as an ideal? (I ask this question due to its use in the third line in the proof in this paper. Does the author mean the ideal to be $\{a\in R:aM \subseteq N\}$? Please help. Thanks in advance
2026-04-22 07:28:15.1776842895
$M$ be a finitely generated $R$-module , and $N$ be a submodule of $M$ ; is it possible to have a meaning for $Ann(M)/N$ as an ideal?
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It seems that the true statement is $Ann (M/N)=p,$ not $Ann (M)/N=p.$ Assume that
Now $a\notin p$ means $a(M/N)\neq 0$, which means $aM\nsubseteq N$. So $N+aM$ strictly contains $N$. (Similar argument is true for $b$ instead of $a$.) Hence $N+aM$ and $N+bM$ are both finitely generated.