annihilator
Let be a ring, and suppose that is a left -module and a right -module.
Annihilator of a Subset of a Module
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1.
If is a subset of , then we define the left annihilator of in :
If , then so are and for all . Therefore, is a left ideal of .
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2.
If is a subset of , then we define the right annihilator of in :
Like above, it is easy to see that is a right ideal of .
Remark. and may also be written as and respectively, if we want to emphasize .
Annihilator of a Subset of a Ring
-
1.
If is a subset of , then we define the right annihilator of in :
If , then so are and for all . Therefore, is a left -submodule of .
-
2.
If is a subset of , then we define the left annihilator of in :
Similarly, it can be easily seen that is a right -submodule of .
Title | annihilator |
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Canonical name | Annihilator |
Date of creation | 2013-03-22 12:01:32 |
Last modified on | 2013-03-22 12:01:32 |
Owner | antizeus (11) |
Last modified by | antizeus (11) |
Numerical id | 8 |
Author | antizeus (11) |
Entry type | Definition |
Classification | msc 16D10 |
Synonym | left annihilator |
Synonym | right annihilator |
Related topic | JacobsonRadical |