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  "body": "\n## Summary\n\nA left primitive ring is a ring that has a faithful simple left module.\nBy the [[Jacobson density theorem|EK0T7FRKCY]], such a ring acts densely on a vector space over a division ring.\nSo it is close to a full ring of linear maps.\nEvery left primitive ring is [[prime|Z9TJSZZYG1]], and a commutative ring is primitive exactly when it is a field.\nPrimitive rings are the building blocks of Jacobson's structure theory: every [[semiprimitive ring|1B2DBKV1C9]] is a subdirect product of primitive rings.\n\n## Explanations\n\n### L1 human\n\nA module is a set on which the ring acts by something like linear maps.\nA ring is primitive if it can act on one irreducible module without losing information, that is, with no nonzero element acting as zero.\nThe typical example is the ring of all linear maps of a vector space $V$, acting on $V$ itself.\nThe integers are not primitive: each simple $\\Z$-module $\\Z/p\\Z$ is killed by $p$.\n",
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