Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or. A module is called a serial module if it is a direct sum of uniserial modules. A ring R is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.
An easy motivational example is the quotient ring for any integer. This ring is always serial, and is uniserial when n is a prime power.
The term uniserial has been used differently from the above definition: for clarification [|see this section].
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. Drozd, D. Eisenbud, A. Facchini, A.W. Goldie, Phillip Griffith, I. Kaplansky, V.V Kirichenko, G. Köthe, H. Kuppisch, I. Murase, T. Nakayama, P. Příhoda, G. Puninski, and R. Warfield. References for each author can be found in and.
Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a ring with unity, and each module is unital.
Properties of uniserial and serial rings and modules
It is immediate that in a uniserial R-module M, all submodules except M and 0 are simultaneously essential and superfluous. If M has a maximal submodule, then M is a local module. M is also clearly a uniform module and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of M can be generated by a single element, and so M is a Bézout module.It is known that the endomorphism ring EndR is a semilocal ring which is very close to a local ring in the sense that EndR has at most two maximal right ideals. If M is required to be Artinian or Noetherian, then EndR is a local ring.
Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are right Bézout rings. A right serial ring R necessarily factors in the form where each ei is an idempotent element and eiR is a local, uniserial module. This indicates that R is also a semiperfect ring, which is a stronger condition than being a semilocal ring.
Köthe showed that the modules of Artinian principal ideal rings are direct sums of cyclic submodules. Later, Cohen and Kaplansky determined that a commutative ring R has this property for its modules if and only if R is an Artinian principal ideal ring. 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. If additionally the ring is assumed to be Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial.
Being right serial is preserved under direct products of rings and modules, and preserved under quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of finite direct sums of uniserial modules are serial modules.
It has been verified that Jacobson's conjecture holds in Noetherian serial rings.
Examples
Any simple module is trivially uniserial, and likewise semisimple modules are serial modules.Many examples of serial rings can be gleaned from the structure sections above. Every valuation ring is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by semisimple rings.
More exotic examples include the upper triangular matrices over a division ring Tn, and the group ring for some finite field of prime characteristic p and group G having a cyclic normal p-Sylow subgroup.
Structure
This section will deal mainly with Noetherian serial rings and their subclass, Artinian serial rings. In general, rings are first broken down into indecomposable rings. Once the structure of 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 is nonzero. This matrix ring is a subring of Mn for some n, and consists of matrices with entries from V on and above the diagonal, and entries from J 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. A complete description of structure in the case of a circle quiver is beyond the scope of this article, but the complete description 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 m=m, 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 R. Then the direct sums and are isomorphic R-modules if and only if n=t and there exist two permutations and of 1,2,...,n such that and 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. The weak form of the Krull-Schmidt Theorem holds not only for uniserial modules, but also for several other classes of modules