Definition:Annihilator of Ring

Definition
Let $\left({R, +, \times}\right)$ be a ring or, more usually, a field.

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.