A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen, P.M. Cohn, Yu.
Goldie, Phillip Griffith, I. Kaplansky, V.V Kirichenko, G. Köthe, H. Kuppisch, I. Murase, T. Nakayama, P. Příhoda, G. Puninski, and R.
[1] Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial, Artinian, Noetherian) then it is assumed the condition holds on both the left and right.
It is immediate that in a uniserial R-module M, all submodules except M and 0 are simultaneously essential and superfluous.
If M is assumed to be Artinian or Noetherian, then EndR M is a local ring.
where each ei is an idempotent element and eiR is a local, uniserial module.
Köthe showed that the modules of Artinian principal ideal rings (which are a special case of serial rings) are direct sums of cyclic submodules.
Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and Warfield: it states that every finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial).
Being uniserial is preserved for quotients of rings and modules, but never for products.
[2] It has been verified that Jacobson's conjecture holds in Noetherian serial rings.
Many examples of serial rings can be gleaned from the structure sections above.
for some finite field of prime characteristic p and group G having a cyclic normal p-Sylow subgroup.
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 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.
It turns out that the quiver structure for a basic, indecomposable, Artinian serial ring is always a circle or a line.
A complete description of structure in the case of a circle quiver is beyond the scope of this article, but can be found in (Puninski 2002).
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.
The dual notion can be defined: the modules are said to have the same epigeny class, denoted
Let U1, ..., Un, V1, ..., Vt be n + t non-zero uniserial right modules over a ring R. Then the direct sums
for every i = 1, 2, ..., n. This result, due to Facchini, has been extended to infinite direct sums of uniserial modules by Příhoda in 2006.
This extension involves the so-called quasismall uniserial modules.
These modules were defined by Nguyen Viet Dung and Facchini, and their existence was proved by Puninski.
This latter term alludes to valuation rings, which are by definition commutative, uniserial domains.
In the 1930s, Gottfried Köthe and Keizo Asano introduced the term Einreihig (literally "one-series") during investigations of rings over which all modules are direct sums of cyclic submodules.
[5] For this reason, uniserial was used to mean "Artinian principal ideal ring" even as recently as the 1970s.
Because of this historical precedent, some authors include the Artinian condition or finite composition length condition in their definitions of uniserial modules and rings.
Expanding on Köthe's work, Tadashi Nakayama used the term generalized uniserial ring[6] to refer to an Artinian serial ring.
Artinian serial rings are sometimes called Nakayama algebras, and they have a well-developed module theory.
Warfield used the term homogeneously serial module for a serial module with the additional property that for any two finitely generated submodules A and B,