annihilator

Let $R$ be a ring, and suppose that $M$ is a left $R$-module and $N$ a right $R$-module.

Annihilator of a Subset of a Module

1. If $X$ is a subset of $M$,then we define the left annihilator of $X$ in $R$:

   $$\mathrm{l}.\mathrm{ann}(X)=\{r\in R\mid rx=0\text{ for all }x\in X\}.$$

   If $a,b\in \mathrm{l}.\mathrm{ann}(X)$, then so are $a-b$ and $ra$ for all $r\in R$. Therefore, $\mathrm{l}.\mathrm{ann}(X)$ is a left ideal of $R$.

2. If $Y$ is a subset of $N$,then we define the right annihilator of $Y$ in $R$:

   $$\mathrm{r}.\mathrm{ann}(Y)=\{r\in R\mid yr=0\text{ for all }y\in Y\}.$$

   Like above, it is easy to see that $\mathrm{r}.\mathrm{ann}(Y)$ is a right ideal of $R$.

Remark. $\mathrm{l}.\mathrm{ann}(X)$ and $\mathrm{r}.\mathrm{ann}(Y)$ may also be written as $\mathrm{l}.{\mathrm{ann}}_R(X)$ and $\mathrm{r}.{\mathrm{ann}}_R(Y)$ respectively, if we want to emphasize $R$.

Annihilator of a Subset of a Ring

1. If $Z$ is a subset of $R$,then we define the right annihilator of $Z$ in $M$:

   $$\mathrm{r}.{\mathrm{ann}}_M(Z)=\{m\in M\mid zm=0\text{ for all }z\in Z\}.$$

   If $m,n\in \mathrm{r}.{\mathrm{ann}}_M(Z)$, then so are $m-n$ and $rm$ for all $r\in R$. Therefore, $\mathrm{r}.{\mathrm{ann}}_M(Z)$ is a left $R$-submodule of $M$.

2. If $Z$ is a subset of $R$,then we define the left annihilator of $Z$ in $N$:

   $$\mathrm{l}.{\mathrm{ann}}_N(Z)=\{n\in N\mid nz=0\text{ for all }z\in Z\}.$$

   Similarly, it can be easily seen that $\mathrm{l}.{\mathrm{ann}}_N(Z)$ is a right $R$-submodule of $N$.