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 R {\displaystyle R} is a commutative ring and I {\displaystyle I} is an ideal of R {\displaystyle R} , we can consider the quotient ring R / I {\displaystyle R/I} to be an R {\displaystyle R} -module. Then, the annihilator of R / I {\displaystyle R/I} is the ideal I {\displaystyle I} , since all of the i ∈ I {\displaystyle i\in I} act via the zero map on R / I {\displaystyle R/I} .

Source: Wikipedia — Annihilator (ring theory) (CC BY-SA 4.0)