annihilator is an ideal

The right annihilator of a right $R$ -module $M_{R}$ in $R$ is an ideal.

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.

Title	annihilator is an ideal
Canonical name	AnnihilatorIsAnIdeal
Date of creation	2013-03-22 12:50:27
Last modified on	2013-03-22 12:50:27
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	10
Author	yark (2760)
Entry type	Theorem
Classification	msc 16D10
Classification	msc 16D25