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annihilator (Definition)

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 $N$ .




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See Also: Jacobson radical

Other names:  left annihilator, right annihilator

Attachments:
annihilator is an ideal (Theorem) by yark
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Cross-references: right ideal, easy to see, left ideal, subset, right, ring
There are 10 references to this entry.

This is version 4 of annihilator, born on 2001-11-24, modified 2008-10-05.
Object id is 996, canonical name is Annihilator.
Accessed 7856 times total.

Classification:
AMS MSC16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)

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