Definition:Annihilator

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Definition

Let $R$ be a commutative ring.

Let $M$ and $N$ be modules over $R$.

Let $B : M \times N \to R$ be a bilinear mapping.


The annihilator of $D \subseteq M$, denoted $\map {\operatorname {Ann}_N} D$ is the set:

$\set {n \in N : \forall d \in D: \map B {d, n} = 0}$


Special Cases

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 $D = R$:

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


Also see

  • Results about annihilators can be found here.


Linguistic Note

The word annihilator calls to mind a force of destruction which removes something from existence.

In fact, the word is a compound construct based on the Latin nihil, which means nothing.

Thus annihilator can be seen to mean, literally, an entity which causes (something) to become nothing.


The pronunciation of annihilator is something like an-nile-a-tor.