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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA [[two-sided ideal|0DDA6CY7GJ]] $P$ of $R$ is *prime* if the following hold.\n\n1. $P \\ne R$.\n2. For all two-sided ideals $A, B$ of $R$, if $AB \\subseteq P$, then $A \\subseteq P$ or $B \\subseteq P$.\n\nHere $AB$ is the ideal generated by the products $ab$ with $a \\in A$ and $b \\in B$.\n\n## Remarks\n\nEquivalently, $P \\ne R$ and, for all $a, b \\in R$, $aRb \\subseteq P$ implies $a \\in P$ or $b \\in P$; and $P$ is prime if and only if $R/P$ is a [[prime ring|Z9TJSZZYG1]].\nExample: the zero ideal of $M_2(\\Q)$ is prime, since the only two-sided ideals are $0$ and $M_2(\\Q)$, and $M_2(\\Q) \\cdot M_2(\\Q) \\ne 0$.\nNon-example: the zero ideal of $\\Q \\times \\Q$ fails clause 2 with $A = \\Q \\times 0$ and $B = 0 \\times \\Q$, since $AB = 0$.\nA *completely prime* ideal is one with $ab \\in P \\Rightarrow a \\in P$ or $b \\in P$; it is prime, but the zero ideal of $M_2(\\Q)$ shows the converse fails.\nFor a commutative ring both notions agree with the usual [[prime ideal|00MHHZ9SNE]].\n",
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