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


 * $ \left\{ n \in N : B(d,n) = 0,\ \forall d \in D\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 $B : M \times G \to R :(m,n) \mapsto n(m)$.

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

In this case the annihilator of $L$ is:


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

Annihilator as ring elements
Let $N = R$.

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

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


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