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  "body": "\n## Summary\n\nThe endomorphism ring of a module is the set of all module homomorphisms from the module to itself, with pointwise addition and composition as multiplication.\nIt is the main source of noncommutative rings: every ring is a subring of the endomorphism ring of its own additive group.\nProperties of a module are often read off its endomorphism ring.\nFor example, a nonzero module is indecomposable exactly when its endomorphism ring has no idempotents other than $0$ and $1$.\nFor a vector space it is the [[algebra of endomorphisms|0E3X2G1ZQB]], and for $K^n$ it is the [[matrix ring|0FHVPRAE60]] $M_n(K)$.\n\n## Explanations\n\n### L1 human\n\nThe structure-preserving maps from an object to itself can be added and composed.\nComposition plays the role of multiplication, and the identity map plays the role of $1$.\nFor the integers as an abelian group, every such map is multiplication by a fixed integer, so the endomorphism ring is $\\Z$ again.\nFor the plane $\\R^2$ as a real vector space, the endomorphisms are the linear maps, and they form the ring of $2 \\times 2$ real matrices.\n",
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