Definition:Annihilator of Ring

Definition

Let $\struct {R, +, \times}$ be a ring or, more usually, a field.

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 $D = R$:

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

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.

From Annihilator of Ring Always Contains Zero, we have that $0 \in \map {\mathrm {Ann} } R$ whatever the ring $R$ is.

$R$ is said to have a trivial annihilator if and only if its annihilator $\map {\mathrm {Ann} } R$ consists only of the integer $0$.

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.