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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].\nAn object $S$ of $\\mathcal{A}$ is *simple* if $S$ is not a zero object and its only [[subobjects|FS1FKAE89H]] are $0$ and $S$.\nLet $S$ and $T$ be simple objects and let $f \\colon S \\to T$ be a morphism.\nThen $f$ is either the [[zero morphism|YF9F9EG6M3]] or an isomorphism.\nIn particular, the endomorphism ring $\\End(S)$ is a [[division ring|0EA5BTA0CX]].\n\n## Remarks\n\nThe [[kernel|S417YNEJ6M]] of $f$ is a subobject of $S$, so it is $0$ or $S$.\nThe image of $f$ is a subobject of $T$, so it is $0$ or $T$.\nIf $f \\ne 0$, then the kernel is not $S$ and the image is not $0$.\nSo $f$ is a [[monomorphism|Y2NP1JFR86]] and an [[epimorphism|RVXDSY04KG]], and in an abelian category such a morphism is an isomorphism.\nFor $\\End(S)$, every nonzero element is invertible, and $1 \\ne 0$ because $S$ is not zero.\nIn the category of modules over a ring, the simple objects are the simple modules, and one gets the classical Schur's lemma.\nThe simplicity hypothesis is needed: multiplication by $2$ on the $\\Z$-module $\\Z$ is nonzero and not an isomorphism, and $\\Z$ is not simple since $2\\Z$ is a proper nonzero submodule.\n\n## Explanations\n\n### L1 human\n\nA simple object is one that cannot be cut into smaller nonzero pieces.\nBetween two such objects, a map either kills everything or matches them up perfectly.\nThere is nothing in between, because the kernel and the image of the map must each be all or nothing.\n\n### L2 human\n\nThe lemma is the first step in studying objects of finite length, which are built from simple objects by extensions as in the Jordan–Hölder theorem.\nIt shows that $\\Hom(S, T) = 0$ for non-isomorphic simple objects and that $\\End(S)$ is a division ring.\nIn representation theory over an algebraically closed field $k$, a further argument with eigenvalues gives $\\End(S) = k$ for finite-dimensional simple representations; that sharper form needs the extra hypotheses.\n",
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