these rings are known, the decomposable rings are direct products of the indecomposable ones. Also, for semiperfect rings such as serial rings, the basic ring is Morita equivalent to the original ring. Thus if R is a serial ring with basic ring B, and the structure of B is known, the theory of Morita equivalence gives that where P is some finitely generated progenerator B. This is why the results are phrased in terms of indecomposable, basic rings. In 1975, Kirichenko and Warfield independently and simultaneously published analyses of the structure of Noetherian, non-Artinian serial rings. The results were the same however the methods they used were very different from each other. The study of hereditary, Noetherian, prime rings, as well as quivers defined on serial rings were important tools. The core result states that a right Noetherian, non-Artinian, basic, indecomposable serial ring can be described as a type of matrix ring over a Noetherian, uniserial domain V, whose Jacobson radical J(V) is nonzero. This matrix ring is a subring of Mn(V) for some n, and consists of matrices with entries from V on and above the diagonal, and entries from J(V) below. Artinian serial ring structure is classified in cases depending on the quiver structure. It turns out that the quiver structure for a basic, indecomposable, Artinian serial ring is always a circle or a line. In the case of the line quiver, the ring is isomorphic to the upper triangular matrices over a division ring (note the similarity to the structure of Noetherian serial rings in the preceding paragraph). A complete description of structure in the case of a circle quiver is beyond the scope of this article, but can be found in . To paraphrase the result as it appears there: A basic Artinian serial ring whose quiver is a circle is a homomorphic image of a "blow-up" of a basic, indecomposable, serial quasi-Frobenius ring. A decomposition uniqueness property Two modules U and V are said to have the same monogeny class, denoted , if there exists a monomorphism and a monomorphism . The dual notion can be defined: the modules are said to have the same epigeny class, denoted , if there exists an epimorphism and an epimorphism . The following weak form of the Krull-Schmidt theorem holds. Let U1, ..., Un, V1, ..., Vt be n + t non-zero uniserial right modules over a ring