 Annihilator (ring theory)

For a subset $X$ of a left module $M$ over a ring $R$, its left annihilator isIt is a left ideal. If $X$ is a submodule, it is a two-sided ideal, because $r(ax)=0$ for every $a\in R$ and $x\in X$. In particular the annihilator of a module measures which scalars are invisible to the entire action; a faithful module has zero annihilator.

For the integer module $\mathbb{Z}/n\mathbb{Z}$, its annihilator is $n\mathbb{Z}$. For a single vector in a noncommutative module, its annihilator need only be a left ideal. The right annihilator instead records elements killed by multiplication from the other side.

In ring theory, the term "annihilator" refers to a specific concept associated with modules over rings, though it can also be extended to other algebraic structures.