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.

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