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 $\operatorname{Ann}_N \left({D}\right)$ is the set:

$\left\{{n \in N : \forall d \in D: B \left({d, n}\right) = 0}\right\}$


Special Cases

Various definitions of the annihilator can be found in the literature, including:


Annihilator on Algebraic Dual

Let $R$ be a commutative ring.

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 $M$, denoted $M^\circ$, is defined as:

$\ M^\circ := \left\{{t' \in G^*: \forall x \in M: t' \left({x}\right) = 0}\right\}$


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 $\left({R, +, \times}\right)$:


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

$B: R \times \Z: \left({r, n}\right) \mapsto n \cdot r$

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

$n \cdot r = r + r + \cdots \left({n}\right) \cdots r$

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

$B \left({r, n}\right) = n \cdot r$


Let $D \subseteq R$.

Then the annihilator of $D$ is defined as:

$\operatorname{Ann} \left({D}\right) = \left\{{n \in \Z: \forall d \in D: n \cdot d = 0_R}\right\}$

or, when $D = R$:

$\operatorname{Ann} \left({R}\right) = \left\{{n \in \Z: \forall r \in R: n \cdot r = 0_R}\right\}$


It is seen to be, therefore, the set of all integers whose integral multiples, with respect to the elements of a ring or a field, are all equal to the zero of that ring or field.


Trivial Annihilator

From Annihilator of Ring Always Contains Zero, we have that $0 \in \operatorname{Ann} \left({R}\right)$ whatever the ring $R$ is.

$R$ is said to have a trivial annihilator if and only if its annihilator $\operatorname{Ann} \left({R}\right)$ consists only of the integer $0$.


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.