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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity.\n$R$ is *left primitive* if there is a [[simple|G00C22AEDH]] left $R$-[[module|09VDS2W1E8]] $M$ that is [[faithful|TB40Y3SNSX]], that is, $rM = 0$ implies $r = 0$ for $r \\in R$.\n*Right primitive* rings are defined in the same way with right modules.\n\n## Remarks\n\nExample: let $V$ be a vector space over a field $k$ and $R = \\End_k(V)$.\n$V$ is a simple $R$-module, since any nonzero vector can be sent to any vector by a linear map.\nIt is faithful, since a linear map that kills every vector is $0$.\nNon-example: $\\Z$ is not primitive, since its simple modules are the groups $\\Z/p\\Z$, and $p$ kills $\\Z/p\\Z$; no simple module is faithful.\nA commutative ring is primitive if and only if it is a field.\nEvery simple ring with identity is left and right primitive.\nA left Artinian ring is left primitive if and only if it is isomorphic to $M_n(D)$ for a division ring $D$ and an integer $n \\ge 1$.\nThere are left primitive rings that are not right primitive.\n",
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  "name": "Primitive ring",
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