$IM$ not finitely generated , $J \subseteq I$, $JM$ finitely generated; is there some $a\in I$ such that $JM\subsetneq\langle a,J\rangle M$?

Question

Let $R$ be a commutative ring with unity, $M$ be an $R$-module, $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that $JM$ is a finitely generated submodule. Then does there exist $a\in I$ such that $JM \subsetneq \langle a,J\rangle M$? I can only figure out that $JM$ must be a proper submodule of $IM$.

Answer 1

Since $JM$ is finitely generated and $IM$ isn't, clearly $JM$ is a proper submodule of $IM$. Let $x\in IM\setminus JM$. Then $$ x=a_1y_1+a_2y_2+\dots+a_ny_n $$ for some $a_1,a_2,\dots,a_n\in I$, $y_1,y_2,\dots,y_n\in M$. Since $x\notin JM$, there is some $i$ such that $a_iy_i\notin JM$. Obviously $a_iy_i\in\langle a_i,J\rangle M$.