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annihilator (Definition)

Let $ R$ be a ring.

Suppose that $ M$ is a left $ R$-module.

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 $ 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 $\displaystyle z \in Z \}.$

Suppose that $ N$ is a right $ R$-module.

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 $\displaystyle y \in Y \}.$

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 $\displaystyle z \in Z \}.$



"annihilator" is owned by antizeus.
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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, subset, ring
There are 11 references to this entry.

This is version 3 of annihilator, born on 2001-11-24, modified 2003-09-20.
Object id is 996, canonical name is Annihilator.
Accessed 6501 times total.

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