Nakayama algebra

Remarks

Example: $A = K[x]/(x^n)$ with $n \ge 1$. The 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. Example: the algebra of upper triangular $n \times n$ matrices over $K$ is a Nakayama algebra; its indecomposable projective left modules are its columns. Non-example: $A = K[x, y]/(x, y)^2$, with basis $1, x, y$. The module $A$ is indecomposable and projective, but $Kx$ and $Ky$ are submodules neither of which contains the other, so $A$ is not uniserial. Over a Nakayama algebra every indecomposable module is uniserial and is a quotient of an indecomposable projective module. In the language of rings, a Nakayama algebra is a finite-dimensional algebra that is a serial ring on both sides.