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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]] with identity and let $U$ be a [[simple|G00C22AEDH]] left $R$-module.\nLet $D = \\End_R(U)$ be its [[endomorphism ring|AWSBZWWMG9]], which is a [[division ring|0WZXK05ME7]], and view $U$ as a vector space over $D$.\nLet $u_1, \\ldots, u_n \\in U$ be linearly independent over $D$, and let $v_1, \\ldots, v_n \\in U$ be arbitrary.\nThen there is $r \\in R$ with $r u_i = v_i$ for $i = 1, \\ldots, n$.\n\n## Remarks\n\n$D$ is a division ring by [[Schur's lemma|TRMYSF91EV]].\nThe proof is by induction on $n$; the key step shows that if $W \\subseteq U$ is a finite-dimensional $D$-subspace and $u \\notin W$, then some $r \\in R$ kills $W$ but not $u$.\nExample: let $R = M_n(k)$ for a field $k$ and $U = k^n$, so that $D = k$.\nThe theorem says that any $n$ linearly independent vectors can be sent to any $n$ vectors by a matrix.\nIf $U$ is finite-dimensional over $D$, the map $R \\to \\End_D(U)$ is surjective; when $R$ is a finite-dimensional algebra over an algebraically closed field this gives Burnside's theorem.\n\n## Explanations\n\n### L1 human\n\nA ring acting irreducibly on a module acts as richly as possible.\nOn any finite set of independent vectors, it can do whatever a linear map can do.\nSo a primitive ring looks like a ring of linear maps that is \"dense\" in the full ring of linear maps.\n\n### L2 human\n\nThe theorem is the main tool for the structure of [[primitive rings|8F6EHDVJ62]]: a left primitive ring is isomorphic to a dense subring of $\\End_D(U)$.\nIf in addition $R$ is left Artinian, the subring is all of $\\End_D(U)$ with $U$ finite-dimensional, so $R \\cong M_n(D)$.\nThis is the route to [[Wedderburn's theorem on simple Artinian rings|YJM07BT0KK]] and to the [[Artin–Wedderburn theorem|R1F271180N]].\n",
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