Definition:Annihilator/Special Cases

Special Cases of Annihilators

Various special cases of annihilators can be found in the literature, including:

Annihilator on Algebraic Dual

Let $R$ be a commutative ring with unity.

Let $G$ be a module over $R$.

Let $G^*$ be the algebraic dual of $G$.

Let $M$ be a submodule of $G$.

The annihilator of $M$ is denoted and defined as:

$M^\circ := \set {t \in G^*: \forall x \in M: \map t x = 0}$

Annihilator of Subspace of Banach Space

Let $X$ be a Banach space.

Let $M$ be a vector subspace of $X$.

Let $X^\ast$ be the normed dual space of $X$.

We define the annihilator $M^\bot$ by:

$M^\bot = \set {g \in X^\ast : \map g x = 0 \text { for all } x \in M}$

Annihilator of Subspace of Normed Dual Space

Let $X$ be a Banach space.

Let $X^\ast$ be the normed dual space of $X$.

Let $N$ be a vector subspace of $X^\ast$.

We define the annihilator ${}^\bot N$ by:

${}^\bot N = \set {x \in X : \map g x = 0 \text { for all } g \in N}$

Annihilator as Integral Multiples of Ring Elements

A common instance is when $N$ is the ring of integers $\Z$ and $M$ is a ring or a field $\struct {R, +, \times}$:

Let $B: R \times \Z$ be a bilinear mapping defined as:

$B: R \times \Z: \tuple {r, n} \mapsto n \cdot r$

where $n \cdot r$ defined as an integral multiple of $r$:

$n \cdot r = r + r + \cdots \paren n \cdots r$

Note the change of order of $r$ and $n$:

$\map B {r, n} = n \cdot r$

Let $D \subseteq R$ be a subring of $R$.

Then the annihilator of $D$ is defined as:

$\map {\mathrm {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$

or, when $D = R$:

$\map {\mathrm {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$