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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity.\nA [[two-sided ideal|0DDA6CY7GJ]] $P$ of $R$ is a *left primitive ideal* if there is a [[simple|G00C22AEDH]] left $R$-[[module|09VDS2W1E8]] $M$ with\n$$P = \\operatorname{ann}_R(M) = \\set{r \\in R : rm = 0 \\text{ for all } m \\in M}.$$\n\n## Remarks\n\nEquivalently, $P$ is left primitive if and only if the quotient ring $R/P$ is a left [[primitive ring|8F6EHDVJ62]].\nExample: in $\\Z$, for a prime $p$ the ideal $p\\Z$ is the annihilator of the simple module $\\Z/p\\Z$, so it is primitive.\nNon-example: the zero ideal of $\\Z$ is prime but not primitive, since $\\Z/0 \\cong \\Z$ is not a field.\nIn a commutative ring the primitive ideals are exactly the maximal ideals.\nThe [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is the intersection of all left primitive ideals of $R$.\nRight primitive ideals are defined with right modules; the two notions can differ.\n",
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