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