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  "body": "\n## Definition\n\nLet $R$ be a ring with identity element $1$.\nA *left module* over $R$ is an abelian group $(M, +)$ together with a map $R \\times M \\to M$, $(r, x) \\mapsto rx$, such that, for all $r, s \\in R$ and $x, y \\in M$:\n\n1. $r(x + y) = rx + ry$.\n2. $(r + s)x = rx + sx$.\n3. $(rs)x = r(sx)$.\n4. $1x = x$.\n\n## Remarks\n\nIf $R$ is a field, then an $R$-module is the same as a [[vector space|000AYPA9AJ]] over $R$.\nEach abelian group is a module over the ring $\\Z$, with $nx = x + \\cdots + x$ ($n$ terms) for $n \u003e 0$.\nThe ring $R$ is a module over itself, and $R^n$ is a module with the operations on coordinates.\nRight modules use a map $M \\times R \\to M$; over a commutative ring, the two notions are the same.\n\n## Explanations\n\n### L1 human\n\nA module is a vector space in which the scalars come from a ring instead of a field.\nWe cannot divide by scalars in general, so many facts about vector spaces fail.\nFor example, a module need not have a basis.\n",
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