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'annihilator'
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| Title of object: |
annihilator |
| Canonical Name: |
Annihilator |
| Type: |
Definition |
| Created on: |
2001-11-24 00:07:03 |
| Modified on: |
2003-09-20 21:30:28 |
| Classification: |
msc:16D10 |
| Synonyms: |
annihilator=left annihilator
annihilator=right annihilator |
Revision comment (for changes between this and next version):
| Changes for correction #14211 ('subscripts'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $R$ be a ring.
Suppose that $M$ is a left $R$-module.
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 $Z$ is a subset of $R$,
then we define the {\it right annihilator} of $Z$ in $M$:
$${\rm r.ann}_M(Z) = \{ m \in M \mid zm = 0 \text{ for all } z \in Z \}.$$
Suppose that $N$ is a right $R$-module.
If $Y$ is a subset of $N$,
then we define the {\it right annihilator} of $Y$ in $R$:
$${\rm r.ann}(Y) = \{ r \in R \mid yr = 0 \text{ for all } y \in Y \}.$$
If $Z$ is a subset of $R$,
then we define the {\it left annihilator} of $Z$ in $N$:
$${\rm l.ann}_N(Z) = \{ n \in N \mid nz = 0 \text{ for all } z \in Z \}.$$ |
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