Uniserial ring

A ring all indecomposable one-sided ideals of which have a unique composition series, and which splits into the direct sum of primary rings. Omission of the latter requirement leads to the definition of a generalized uniserial ring, also called a serial ring. Every generalized uniserial ring is a semi-chain ring both on the left and the right (see Semi-chain module; Semi-chain ring). Every module over a generalized uniserial ring splits into the direct sum of cyclic submodules. A ring is a generalized uniserial ring if and only if all left modules over it are semi-chain modules. An example of a uniserial ring is the ring of upper-triangular matrices over a skew-field.