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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nA *minimal prime ideal* of $R$ is a [[prime ideal|06HC6HK0Z8]] $P$ of $R$ such that no prime ideal $Q$ of $R$ satisfies $Q \\subsetneq P$.\nLet $I$ be an [[ideal|0MYSA6MYNA]] of $R$.\nA *minimal prime over $I$* is a prime ideal $P \\supseteq I$ such that no prime ideal $Q$ satisfies $I \\subseteq Q \\subsetneq P$.\n\n## Remarks\n\nExample: in $R = k[x, y]/(xy)$ with $k$ a field, the ideal $(x)$ is a minimal prime.\nIt is prime because $R/(x) \\cong k[y]$ is a domain.\nA prime $Q \\subseteq (x)$ contains $xy = 0$, so it contains $x$ or $y$, and $y \\notin (x)$ forces $x \\in Q$.\nNon-example: $2\\Z$ is prime in $\\Z$, but it is not minimal, since the prime ideal $(0)$ is strictly smaller.\nBy Zorn's lemma, every prime ideal containing $I$ contains a minimal prime over $I$; the key point is that the intersection of a chain of prime ideals is prime.\nThe minimal primes over $I$ are the minimal primes of $R/I$, pulled back to $R$.\n",
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