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  "body": "\n## Definition\n\nLet $K$ be a field and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $K$.\nA module is *uniserial* if it has a unique [[composition series|0XV2A49X6Q]], equivalently if its submodules are totally ordered by inclusion.\nThe algebra $A$ is a *Nakayama algebra* if every [[indecomposable|E2P29GCD9T]] [[projective|BFFA03GE3V]] left $A$-module and every indecomposable projective right $A$-module is [[uniserial|Z1045A0V29]].\n\n## Remarks\n\nExample: $A = K[x]/(x^n)$ with $n \\ge 1$.\nThe only indecomposable projective module on either side is $A$ itself, and its submodules are the ideals $(x^i)$ for $0 \\le i \\le n$, which form a chain.\nExample: the algebra of upper triangular $n \\times n$ matrices over $K$ is a Nakayama algebra; its indecomposable projective left modules are its columns.\nNon-example: $A = K[x, y]/(x, y)^2$, with basis $1, x, y$.\nThe module $A$ is indecomposable and projective, but $Kx$ and $Ky$ are submodules neither of which contains the other, so $A$ is not uniserial.\nOver a Nakayama algebra every indecomposable module is uniserial and is a quotient of an indecomposable projective module.\nIn the language of rings, a Nakayama algebra is a finite-dimensional algebra that is a [[serial ring|TW3ESQB3GQ]] on both sides.\n",
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