Definition:Annihilator

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 as Linear Forms

Let $N = M^*$ be the algebraic dual of $M$.

Let $B : M \times G \to R : \left({m, n}\right) \mapsto n \left({m}\right)$.

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

In this case the annihilator of $L$ is:

$L^\circ := \left\{{n \in M^*: \forall \ell \in L: n \left({\ell}\right) = 0}\right\}$

Annihilator as Ring Elements

Let $N = R$.

Let $B \left({m, r}\right) = r \cdot m$, where $\cdot$ is the multiplication from the module structure.

In this case, for $D \subseteq M$:

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

Annihilator as Integral Multiples of Ring Elements

The usual case 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.

The trivial annihilator is an annihilator which contains only 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.

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$