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:


 * $\map {\operatorname {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$

or, when $D = R$:
 * $\map {\operatorname {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$

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.

Also see

 * Definition:Annihilator