Annihilator (ring theory)

In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that always give zero when multiplied by each element of S. For example, if ${\displaystyle R}$ is a commutative ring and ${\displaystyle I}$ is an ideal of ${\displaystyle R}$, we can consider the quotient ring ${\displaystyle R/I}$ to be an ${\displaystyle R}$-module. Then, the annihilator of ${\displaystyle R/I}$ is the ideal ${\displaystyle I}$, since all of the ${\displaystyle i\in I}$ act via the zero map on ${\displaystyle R/I}$.

Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator.

The above definition applies also in the case of noncommutative rings, a subset of a left module has a left annihilator, which is a left ideal, and a subset of a right module has a right annihilator, which is a right ideal. If ${\displaystyle S}$ is a module, then the annihilator is always a two-sided ideal, regardless of whether the module is a left module or a right module.