annihilator is an ideal
AnnihilatorIsAnIdeal
2002-07-15 09:03:29
2007-01-18 05:23:25
Theorem
annihilator
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The right annihilator of a right $R$-module $M_R$ in $R$ is an ideal.
{\bf Proof:}\\
By the distributive law for modules, it is easy to see that $\operatorname{r.ann}(M_R)$ is closed under addition and right multiplication.
Now take $x \in \operatorname{r.ann}(M_R)$ and $r \in R$.
Take any $m \in M_R$. Then $mr \in M_R$, but then $(mr)x = 0$ since $x \in \operatorname{r.ann}(M_R)$. So $m(rx)=0$ and $rx \in \operatorname{r.ann}(M_R)$.
An equivalent result holds for left annihilators.