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  "body": "\n## Summary\n\nA serial ring is a [[ring|0B1261B5ED]] that, as a module over itself on each side, splits into a direct sum of [[uniserial modules|Z1045A0V29]].\nSerial rings include [[discrete valuation rings|MGM3CNJW9V]], the rings $\\Z/n\\Z$, and the rings of [[upper triangular matrices|0N4XY7Z3NN]] over a field.\nThe finite-dimensional serial algebras over a field are the [[Nakayama algebras|APRFMTZ10K]].\nTheir module theory is very explicit.\nBy a theorem of Warfield, every finitely presented module over a serial ring is a direct sum of uniserial modules.\n\n## Explanations\n\n### L1 human\n\nFor a serial ring, the ring itself can be cut into a few pieces, and each piece has submodules that line up in a single chain.\nSo the whole ring is a sum of \"ladders\" with no branching.\nThe ring $\\Z/12\\Z$ is serial: it splits as $\\Z/4\\Z \\times \\Z/3\\Z$, and each factor has a chain of ideals.\nThe ring $k[x, y]/(x, y)^2$ over a field $k$ is not serial, because the ideals generated by $x$ and by $y$ cannot be compared.\n",
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