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I'm stuck with the following question in the sense that I admit I cannot come up with a decent solution. Here is the question:

We define a submodule $N$ of the $R$ module $M_R$ to be essential (large) in $M_R$ if, for any submodule $L\leq M_R$, $N\cap L=0$ implies $L=0$.

Then prove the following:

A submodule $N\leq M_R$ is essential in $M_R$ if and only if $\;\forall\; x\in M,\;x\neq 0$, there exists $r\in R$ such that $xr\in N$ and $xr\neq 0$.

Thanks in advance

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    Next time *please* include whatever it is that you got (a non-decent solution?) in your question. The best way for us to help you unstuck yourself is to know *where* you got stuck!2012-01-18

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Suppose $N$ is essential in $M$ and let $x\in M$ be a non-zero element. Then the submodule $xR$ of $M$ is not-zero, so essentiality implies that $xR\cap N\neq0$. This means, precisely, that there is an $r\in R$ such that $xr\in N$ and $xr\neq0$.

Conversely, suppose $N$ is a submodule of $M$ which has that property and let $L\subseteq M$ be a non-zero submodule. If $x\in L$ is a non-zero element, by hypothesis there is an $r\in R$ such that $xr\in N$ and $xr\neq 0$. But then $0\neq xr\in N\cap xR\subseteq N\cap L$, so $N\cap L\neq0$. We thus see that $N$ is essential.

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    oh yes... didn't even think about that... by the way... thank you very much for your kindness. Regards.2012-01-18