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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0310REJJY8]] with [[Jacobson radical|XA0DCNH7WY]] $J(R)$.\nLet $I \\subseteq J(R)$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-[[module|0NZ8JA25WK]].\nIf $IM = M$, then $M = 0$.\n\n## Remarks\n\nProof idea: let $m_1, \\ldots, m_n$ generate $M$ with $n$ minimal, and suppose $n \\ge 1$.\nFrom $IM = M$, write $m_n = a_1 m_1 + \\cdots + a_n m_n$ with $a_i \\in I$.\nThen $(1 - a_n) m_n$ lies in the submodule generated by $m_1, \\ldots, m_{n-1}$.\nSince $a_n \\in J(R)$, the element $1 - a_n$ is a unit, so $m_n$ is not needed, contradicting minimality.\nAnother proof uses the [[determinant trick|A8HRT9H3XA]] to find $a \\in I$ with $(1 + a) M = 0$.\n\nConsequence: if $N \\subseteq M$ is a [[submodule|05WDSN597C]] and $M = N + IM$, then $M = N$; apply the lemma to $M/N$.\nFor a [[local ring|MZ02BKPHJA]] $(R, \\mathfrak m)$ with $I = \\mathfrak m$, this says that elements of $M$ generate $M$ exactly when their images span the vector space $M/\\mathfrak m M$ over $R/\\mathfrak m$.\n\nBoth hypotheses are needed.\nWithout finite generation: over $R = \\Z_{(p)}$ with $I = p\\Z_{(p)} = J(R)$, the module $M = \\Q$ satisfies $pM = M$ but $M \\ne 0$.\nWithout $I \\subseteq J(R)$: over $R = \\Z$ with $I = 2\\Z$, the module $M = \\Z/3\\Z$ satisfies $2M = M$, since $2$ is invertible modulo $3$, but $M \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nOver a local ring, the maximal ideal consists of \"small\" elements.\nNakayama's lemma says that a finitely generated module cannot be made of small multiples of itself unless it is zero.\nIn practice it says that to find generators of a module, it is enough to find generators \"modulo the maximal ideal\", which is a question of linear algebra.\n\n### L2 human\n\nThe lemma is the main tool for passing from the [[residue field|NJ3APQ8BHH]] back to the local ring.\nIt gives the equality between the minimal number of generators of a finitely generated module $M$ over a local ring and $\\dim_k M/\\mathfrak m M$.\nWith $M = \\mathfrak m$, this identifies the [[embedding dimension|DP3KSCRKFP]] with the minimal number of generators of $\\mathfrak m$.\nIt also shows that [[finitely generated projective modules over local rings are free|942FK83S09]], and it enters the proof of the [[Krull intersection theorem|1Q1AH2V41S]].\n",
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