Faithful module (${\mathrm{Ann}}_R(M)=0$)

A module is faithful when its annihilator of a module is zero. Equivalently, the action ring homomorphism $R\to {\mathrm{End}}_{\mathbb{Z}}(M)$ to the endomorphism ring of its additive abelian group is injective: every nonzero scalar acts nontrivially somewhere on $M$. A primitive ring has a faithful simple module.