# 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 $\mathrm{r}.\mathrm{ann}({M}_{R})$ is closed under addition and right multiplication.
Now take $x\in \mathrm{r}.\mathrm{ann}({M}_{R})$ and $r\in R$.

Take any $m\in {M}_{R}$. Then $mr\in {M}_{R}$, but then $(mr)x=0$ since $x\in \mathrm{r}.\mathrm{ann}({M}_{R})$. So $m(rx)=0$ and $rx\in \mathrm{r}.\mathrm{ann}({M}_{R})$.

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 |