Annihilator

left, of a set $X$ in $R$

The set $\mathfrak{Z}_l(X)$ of all elements $y$ in $R$ such that $yX = \{0\}$. Here $R$ is a ring or a semi-group (or, generally, a groupoid) with a zero. The right annihilator of a set $X$ in $R$ is defined in a similar manner as the set $$ \mathfrak{Z}_r(X) = \{ z \in R : Xz = \{0\} \} \ . $$

The set $$ \mathfrak{Z}(X) = \mathfrak{Z}_l(X) \cap \mathfrak{Z}_r(X) $$ is the two-sided annihilator of $X$. In an associative ring (or semi-group) $R$ the left annihilator of an arbitrary set $X$ is a left ideal, and if $X$ is a left ideal of $R$, then $\mathfrak{Z}_l(X)$ is a two-sided ideal of $R$; in the non-associative case these statements are usually not true.

Modules

Let $M$ be a left module over a ring $R$, and $X$ a subset of $M$. The left annihilator of $X$ is $$ \mathfrak{Z}_l(X) = \{ z \in R : Xz = \{0\} \} \ . $$ Again, the left annihilator of an arbitrary set $X$ is a left ideal. The annihilator of an element $x \in M$ is the annihilator of $\{ x \}$. As left $R$-modules we have $$ R/\mathfrak{Z}_l(\{x\}) \cong Rx \ . $$

References

[a1]	N. Bourbaki, "Algebra I" , Springer (1998) ISBN 3-540-64243-9
[a2]	N. Bourbaki, "Algebra II" , Springer (2003) ISBN 3-540-00706-7
[a3]	S. Lang, "Algebra" , Springer (2002) ISBN 0-387-95385-X

Linear spaces

Let $V$ be a vector space over a field $K$ and $V^*$ the dual space of linear functional‎s on $V$. For a subset $X$ of $V$, the annihilator $$ X^\circ = \{ f \in V^* : f(X) = \{0\} \} \ . $$ The annihilator of a general set $X$ is a subspace of $V^*$ and if $\langle X \rangle$ is the subspace of $V$ generated by $X$, then $X^\circ = \langle X \rangle^\circ$.