# 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 $$R/\mathfrak{Z}_l(\{x\}) \cong Rx \ .$$

How to Cite This Entry:
Annihilator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Annihilator&oldid=34143
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article