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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity.\nA right $R$-[[module|0NZ8JA25WK]] is *serial* if it is a direct sum of [[uniserial modules|Z1045A0V29]].\nThe ring $R$ is *right serial* if the right module $R_R$ is serial.\nIt is *left serial* if the left module ${}_R R$ is serial.\nIt is *serial* if it is both right serial and left serial.\n\n## Remarks\n\nExample: let $k$ be a field and $R = k[x]/(x^n)$ with $n \\ge 1$.\nIts ideals are the ideals $(x^i)/(x^n)$ for $0 \\le i \\le n$, and they form a chain.\nSo $R$ is uniserial over itself on both sides, and $R$ is serial.\n\nNon-example: let $R = k[x, y]/(x, y)^2$.\nThis is a local ring, so $R$ has no idempotents other than $0$ and $1$, and $R$ is indecomposable as a module over itself.\nIf it were serial, it would therefore be uniserial.\nBut the ideals $(x)$ and $(y)$ of $R$ are not comparable, so $R$ is not serial.\n\nEquivalently, $R$ is right serial when $R = e_1 R \\oplus \\cdots \\oplus e_m R$ for some orthogonal idempotents $e_1, \\dots, e_m$ with each $e_i R$ uniserial, since a direct summand of $R_R$ has the form $eR$ for an idempotent $e$.\nA finite-dimensional serial algebra over a field is also called a [[Nakayama algebra|APRFMTZ10K]].\n",
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