Given a subset S of a module M over a commutative ring R, then the annihilator of S is Ann_R(S) = {r ∈ R : rs =0 for all s ∈ S}. It is an ideal of R.

Given a subset S of a vector space V, then the annihilator S° is the set of linear functionals f such that f(S) ={0}. Then S° is a subspace of the dual space V*; if S is itself a subspace and V is finite-dimensional, then dimS + dimS° = dimV. The correspondence S ↔ S° is crucial to duality in projective geometry.