Serial module

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 End_R(M) is a semilocal ring which is very close to a local ring in the sense that End_R(M) has at most two maximal right ideals. If M is required to be Artinian or Noetherian, then End_R(M) 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 R = ⊕ i = 1 n e i R where each e_i is an idempotent element and e_iR 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 (which are a special case of serial 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 (and hence is serial). 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 (Příhoda 2004).

It has been verified that Jacobson's conjecture holds in Noetherian serial rings.(Chatters & Hajarnavis 1980)

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 T_n(D), and the group ring F [ G ] 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 R ≅ E n d B ( P ) 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 M_n(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 the complete description can be found in (Puninski 2001). 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 [U]_m=[V]_m, if there exists a monomorphism U → V and a monomorphism V → U . The dual notion can be defined: the modules are said to have the same epigeny class, denoted [ U ] e = [ V ] e , if there exists an epimorphism U → V and an epimorphism V → U .

The following weak form of the Krull-Schmidt theorem holds. Let U₁,... U_n, V₁, ..., V_t be n+t non-zero uniserial right modules over a ring R. Then the direct sums U 1 ⊕ ⋯ ⊕ U n and V 1 ⊕ ⋯ ⊕ V t are isomorphic R-modules if and only if n=t and there exist two permutations σ and τ of 1,2,...,n such that [ U i ] m = [ V σ ( i ) ] m and [ U i ] e = [ V τ ( i ) ] e 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 (biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules.)

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