# 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. 1.

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

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

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

2. 2.

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

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

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

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

## Annihilator of a Subset of a Ring

1. 1.

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

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

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

2. 2.

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

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

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

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