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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$:
    $\displaystyle {\rm l.ann}(X) = \{ r \in R \mid rx = 0$    for all $\displaystyle x \in X \}.$
    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$:
    $\displaystyle {\rm r.ann}(Y) = \{ r \in R \mid yr = 0$    for all $\displaystyle y \in Y \}.$
    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$:
    $\displaystyle {\rm r.ann}_M(Z) = \{ m \in M \mid zm = 0$    for all $\displaystyle z \in Z \}.$
    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$:
    $\displaystyle {\rm l.ann}_N(Z) = \{ n \in N \mid nz = 0$    for all $\displaystyle z \in Z \}.$
    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 11 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 7081 times total.

Classification:
AMS MSC: 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)
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