# Show that $T (S^{−1}M)= S^{−1}(T (M))$ [closed]

Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann_R(m)} \neq 0$, i.e. if $rm = 0$ for some nonzero $r \in R$. Denote the set of torsion elements in $M$ by $T(M)$, called the torsion submodule of $M$. We say that $M$ is torsion-free if $T(M) = 0$.

let $S$ be a m.c.s. in the integral domain $R$. Show that $$T (S^{−1}M)= S^{−1}(T (M)).$$

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Presumably, you mean "multiplicative closed set," but that is not a standard acronym, and if you want a better chance of getting answers, you should spell out terminology rather than assuming people know what your acronym means. –  Thomas Andrews Feb 19 '13 at 18:52
This looks like the same user as Mathproof P, who earlier today had spammed 6 similarly worded imperative questions with the same idiosyncratic notation and subject matter. –  rschwieb Feb 19 '13 at 18:59
I have deleted all the wrong tags. –  Martin Brandenburg Feb 19 '13 at 19:09
Presumably the inclusion $S^{-1}(T(M)) \subseteq T(S^{-1}M)$ is clear and it's the other direction that needs more explanation.
Hint: Say $\frac{m}{s} \in T(S^{-1}M)$. This means there is an $\frac{a}{s'} \in S^{-1}R$ such that $\frac{am}{ss'} = 0$ in $S^{-1}M$. Using the definition of localization what does $\frac{am}{ss'} = 0$ translate to as a statement about elements of $R$ and $M$?