Annihilator (ring theory)
There are two conceptions of mathematics, which are referred to with the word Annihilator (or Annullator ).
Annihilator in the context of forms
The Annullatorraum is a generalization of the orthogonal complement of vector spaces, where the dual space can not be identified by a scalar product with the space itself.
Definition
Be a vector space, the associated dual space and a subset of. Then say
The annihilator of.
Properties of the Annihilators
- Is a subspace of the dual space. Therefore, one also speaks of Annullatorraum.
- Wherein the subspace is generated.
- If, as is.
- Is finite and is a subspace of, the following applies. In this case, and the Bidualraum canonically isomorphic and it is, and has been identified with one another.
Annihilator of a module
There was a ring and a module. Then the annihilator of is
One can also describe the Annihilator as the core structure Figure
The Annihilator is an ideal in.