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  "body": "\n## Summary\n\nA faithful module over a ring $R$ is a [[module|0NZ8JA25WK]] on which no nonzero element of $R$ acts as zero.\nEquivalently, its [[annihilator|HB4XDKJTW8]] is zero, so $R$ embeds in the endomorphism ring of the additive group of the module.\nIt is the module version of a [[faithful action|0P7G9NGG2G]] of a group.\nFaithful simple modules define primitive rings, a central class in the structure theory of rings.\n\n## Explanations\n\n### L1 human\n\nA module is faithful when the ring can be fully seen through its action on the module.\nIn a faithful module, two different ring elements always act in different ways.\nThe integers act faithfully on $\\Z$, since $n \\cdot 1 = n$ is nonzero for $n \\ne 0$.\nThey do not act faithfully on $\\Z/5\\Z$, since $5$ acts as zero there.\n",
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