annihilatorAnnihilator2001-11-24 00:07:032008-10-05 15:10:24Definition\usepackage{amssymb}
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\usepackage{xypic}Let $R$ be a ring, and suppose that $M$ is a left $R$-module and $N$ a right $R$-module.
\subsubsection*{Annihilator of a Subset of a Module}
\begin{enumerate}
\item
If $X$ is a subset of $M$,
then we define the {\it 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 $R$.
\item
If $Y$ is a subset of $N$,
then we define the {\it 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 $R$.
\end{enumerate}
\textbf{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 $R$.
\subsubsection*{Annihilator of a Subset of a Ring}
\begin{enumerate}
\item
If $Z$ is a subset of $R$,
then we define the {\it 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 $M$.
\item
If $Z$ is a subset of $R$,
then we define the {\it 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 $N$.
\end{enumerate}