1
$\begingroup$

I am asked to show that if $M$ is an essential extension of $N$, then $S^{-1}M$ is an essential extension of $S^{-1}N$.

So this is how I approached the problem:

Let $U\neq (0)$ be a submodule of $S^{-1}M$. Then we can write $U = S^{-1}V$ for some submodule $V$ of $M$. Since $U\neq (0)$, $V\neq (0)$. Since $M$ is an essential extension of $N$, and $V$ is non-zero submodule of $M$, $V\cap N\neq (0)$. So choose $x\neq 0$ in $V\cap N$.

My problem is now trying to get a non-zero element of $U\cap S^{-1}N$. I initially thought I could just use $x/1$ but I do not know that this is non-zero. So now I am stuck.

Any suggestions?

  • 0
    No problem. In your case, if $S$ consists of regular elements of $R$, then the result is certainly true, and Goodearl and Jordan treat this case in detail, without the commutativity assumption.. When you get the solution, I'd be interested in seeing how it goes.2012-02-29

0 Answers 0