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  "body": "\n## Example\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nThe *category of left $R$-modules* $R\\text{-}\\mathbf{Mod}$ has the left [[$R$-modules|0NZ8JA25WK]] as objects and the $R$-linear maps as morphisms, with composition of maps.\nA map $f \\colon M \\to N$ is $R$-linear when $f(x + y) = f(x) + f(y)$ and $f(rx) = r f(x)$ for all $x, y \\in M$ and $r \\in R$.\nIt is an [[abelian category|Y9CZ3MTNF1]].\nEach $\\Hom_R(M, N)$ is an abelian group under pointwise addition, and composition is bilinear.\nThe zero module is a zero object, and the direct sum $M \\oplus N$ is both product and coproduct.\nThe kernel of $f \\colon M \\to N$ is the submodule $\\set{x \\in M : f(x) = 0}$ and the cokernel is $N / f(M)$.\nAs in abelian groups, each injective linear map is the kernel of its cokernel and each surjective linear map is the cokernel of its kernel.\n\n## Remarks\n\nFor $R = \\Z$ one gets the [[category of abelian groups|99ZVSM254K]], and for a field $R = k$ the category of vector spaces over $k$.\nThe module $R$ itself is a [[generator|CK08JW32WB]] and a [[projective object|YXMXAGVC1Q]], and the category has [[enough projectives|J2TX4TDPCX]] since every module is a quotient of a free module.\nIt also has enough injective objects, which is the basis for derived functors such as $\\operatorname{Ext}$ and $\\operatorname{Tor}$.\nTwo rings whose module categories are equivalent are called Morita equivalent; for example $R$ and the ring of $n \\times n$ matrices over $R$ are Morita equivalent.\n\n## Explanations\n\n### L1 human\n\nThe category of modules over a ring gathers all modules and all linear maps between them.\nIt behaves much like linear algebra over a field, except that modules need not have bases.\nFor example, over $R = \\Z$ the module $\\Z/2\\Z$ is not free, yet it is the cokernel of the map $\\Z \\to \\Z$ given by multiplication by $2$.\n",
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