Faithful module

Remarks

Equivalently, the ring homomorphism $R \to \End_\Z(M)$ that sends $r$ to the map $m \mapsto rm$ is injective. Example: for a field $K$, the column space $K^n$ is a faithful $M_n(K)$-module. The check: if $A e_j = 0$ for every standard basis vector $e_j$, then every column of $A$ is zero, so $A = 0$. Non-example: for $n \ge 2$, the $\Z$-module $\Z/n\Z$ is not faithful, since $n \ne 0$ lies in its annihilator. Every module $M$ is faithful as a module over $R / \operatorname{ann}_R(M)$.