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  "body": "\n## Definition\n\nLet $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].\nThe module $M$ is *faithful* if its [[annihilator|VQ31D68MDD]]\n$$\\operatorname{ann}_R(M) = \\set{r \\in R : rm = 0 \\text{ for all } m \\in M}$$\nis $\\set{0}$.\n\n## Remarks\n\nEquivalently, the ring homomorphism $R \\to \\End_\\Z(M)$ that sends $r$ to the map $m \\mapsto rm$ is injective.\nExample: for a field $K$, the column space $K^n$ is a faithful $M_n(K)$-module.\nThe check: if $A e_j = 0$ for every standard basis vector $e_j$, then every column of $A$ is zero, so $A = 0$.\nNon-example: for $n \\ge 2$, the $\\Z$-module $\\Z/n\\Z$ is not faithful, since $n \\ne 0$ lies in its annihilator.\nEvery module $M$ is faithful as a module over $R / \\operatorname{ann}_R(M)$.\n",
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