Summability and direct sum of uniserial modules

Let R be an associative ring with 1. A left R module M is uniserial i f the lattice L(M) of its submodules is totally ordered under inclusion. We give an example of a uniserial module M with the property of having two submodules 0 < H < K < M such that M is isomorphic to K/H (we call a module M with this property shrinkable). Then we give an example of a uniserial module M isomorphic to all its nonzero quotients M/N, N<M, and with L(M) isomorphic to ω2+1; this solves a problem of Hirano and Mogami [7]. Finally we show that for uniserial modules the property of being shrinkable is connected to the problem of deciding whether a module, which is both a homomorphic image of a finite direct sum of uniserial modules and a submodule of a finite direct sum of uniserial modules, is a finite direct sum of uniserial modules

In this paper, we study a generalization of $h$-pure submodules as well as some other closely related concepts. Here, we examine the extent of this generalization in several ways. We then use this to give a characterization of the imbedded-complete modules. It is found that imbeddedness can considerably more abundant than $h$-purity on direct sum of uniserial modules.

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules.

According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.

The first three sections of this chapter present the structure of serial rings of Warfield [75]. The main theorem 25.3.4 characterizes when every finitely presented left module over a ring R is a direct sum of uniserial modules: this happens iff R is itself such a direct sum both as right and left module, that is, iff R is serial. (See Section 0 for definitions.) In this case, then for any finitely generated submodule M of a finitely generated projective module P, there are “stacked” decompositions of P and M into direct sums of uniserial modules (see 25.3.3ff). Moreover, any Noetherian serial ring is decomposable into a finite product of Artinian and (semi)prime rings (25.3.5). This is reminiscent of the theorems of Chatters (20.30) for hereditary rings, and Krull-Asano-Goldie 20.37, for principal ideal rings, and, in fact, Robson’s general method (20.35) used to prove these also applies here.KeywordsLeft IdealPrime RingProjective ModuleUniserial ModuleSerial RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

Suppose M is a QTAG-module with a subsocle S such that M/S is a direct sum of uniserial modules. Our global aim here is to investigate an interesting connection between the structure of M/S and the QTAG-module M. Specifically, the condition S=Soc(N) for some h-pure submodules N of M allows M to inherit the structure of M/S.

Weak Krull–Schmidt for Infinite Direct Sums of Uniserial Modules

QF-1 algebras with faithful direct sums of uniserial modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper we investigate the class of QTAG-modules whose every separable module is a direct sum of uniserial modules; such modules are called ω-totally ω-projective. We discuss an interesting characterization of this class and we show that the class of -totally -projective modules contains the class of ω-totally ω-projective modules.

Serial and semidistributive modules and rings

On uniserial modules that are not quasi-small

Equivalence of diagonal matrices over local rings

Recently H. Marubayashi [1,2] and S. Singh [10,11,12] generalized some results of torsion abelian groups for modules over some restricted rings, like bounded Dedekind prime rings, bounded hereditary Noetherian prime rings. Singh [12] introduced the concept of h-purity for a module MR satisfying the following conditions:(I) Every finitely generated submodule of every homomorphic image of M is a direct sum of uniserial modules.

The string almost gentle algebras (SAG-algebras) are studied in this paper. Generalizing the properties of string almost gentle algebras has also given. Let be the string almost gentle algebras, with quiver and admissible ideal of algebra. We show that the radical of string almost gentle algebras can be written as a direct sum of uniserial modules. After that the quiver is constructed from and an extension of the quiver is showed. Also, the quiver which is a quiver of trivial extension of string almost gentle algebras has been studied. Consequently, the relations of algebra have been described. As well as, we will show that the algebra is an extension to string almost gentle algebras. Furthermore, we describe the trivial extension of string almost gentle algebras , we prove and that the trivial extension of is isomorphic to the algebra .