Direct Sum Of Uniserial Modules Research Articles

Imbeddedness and 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.

Direct Summand of Serial 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.

On QTAG-Modules Having All N-High Submodules h-Pure

The paper is concerned with h-pure-N-high submodules of QTAG-modules. Here, we characterize the submodules N of an h-reduced QTAG-module for which all h-pure-N-high submodules are bounded. We also discuss some interesting properties of subsocles and consequently give a characterization of the direct sum of uniserial modules.

Homomorphisms and direct sum of uniserial modules

Homomorphisms and direct sum of uniserial modules

Subsocles and direct sum of uniserial modules

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.

Structures of QTAG-modules determined by their submodules

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 study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

Nice Bases of QTAG-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 universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

On totally projective QTAG-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.

On projective QTAG-modules

In this paper we introduce the new class of QTAG-modules namely $\omega_1$-$(\omega+n)$-projective modules, which is an amalgamation of three important classes of modules: $n$-bounded modules, the direct sum of uniserial modules and the countably generated modules. This class is given many equivalent characterizations including being the smallest class containing the $(\omega+n)$-projective modules that is closed with respect to $\omega_1$-bijective homomorphisms.

On countably generated QTAG-modules

A right module M over an associative ring with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. The main aim of this paper is to investigate the direct sums of countably generated QTAG-modules and obtain some significant results related to these modules.

Different Characterizations of Large Submodules of QTAG-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. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule L of M is large in M if L+B=M, for every basic submodule B of M. The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on.

ON DIFFERENT NOTIONS OF TRANSITIVITY FOR QTAG-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. Recently, the authors introduced the classes of QTAG-modules namely as socle-regular and strongly socle-regular QTAG-modules which properly contain the classes of transitive and fully transitive QTAG-modules respectively. Here we define strongly and quasi transitivities and study the inter relations between various type of transitivities.

The Seriality of the Group Ring of a Finite Group Depends Only on the Characteristic of the Field

Let F be a field, and let G be a finite group. In this note, we prove that the seriality of the group ring FG depends only on the characteristic of F . For instance, the ring FpG is serial for a prime field with p elements if and only if the ring FpG is serial, where Fp denotes the algebraic closure of Fp. This solves one problem from the book [3]. We briefly recall basic definitions. A module M is said to be uniserial if its lattice of submodules is a chain; M is serial if it is a direct sum of uniserial modules. The ring R is said to be serial if R is a serial right and serial left R-module. It is well known that the ring R is serial if and only if there exists a complete orthogonal collection of indecomposable idempotents e1, . . . , en ∈ R such that each right module eiR is uniserial, and also each left module Rei is uniserial. Some authors call serial rings Nakayma rings (which is quite justified); therefore serial finite-dimensional algebras over a field are called Nakayama algebras (see [3, Chap. 9]). Let F be an arbitrary field, and let G be a finite group. Of interest to us is the question as to when the group ring FG is serial. Baba and Oshiro suggested the following problem.

Direct products of modules whose endomorphism rings have at most two maximal ideals

Monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules. In this paper, we show that they also turn out to be useful in the study of direct products.

On projection-invariant submodules of QTAG-modules

Abstract 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. Here we study projection-invariant submodule of QTAG -module. A submodule N of a QTAG -module M is said to be projection-invariant in M if f ( N )⊆ N , for all idempotent endomorphisms f in End( M ). Clearly, fully invariant submodules are projection-invariant. Mehdi et. al. characterized fully invariant submodules and characteristic submodules with the help of their socles. Here we investigate the socles of projection-invariant submodules of QTAG -modules.

On n -layered Q T A G -modules

A module M over an associative ring with unity is a Q T A G -module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating concepts related to these modules. Here we introduce the notion of n-layered Q T A G -modules and discuss some interesting properties of these modules. We show that a Q T A G -module M is n-layered if and only if M / N is an n-layered module, whenever N is a finitely generated submodule of M and n ≥ 1 is an integer.

On submodules of QTAG-modules

A module M over an associative ring 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 concepts related to these modules out of which summability and h -purity are very significant. The concepts of summable and σ -summable modules were defined by Naji but there were some questions which remained unanswered. This motivated us to define weakly n-summable and nlayered QTAG-modules with the help of their nice modules. We also study α -modules and ℵ 1 -U -modules with some of their interesting properties. Keywords : QTAG -modules, weakly n-summable modules, n-layered QTAG -modules, Nice submodules.

On elongations of QTAG-modules

Abstract Mehdi studied (ω+k)-projective QTAG-modules with the help of their submodules contained in H k (M) (the submodule generated by the elements of exponents at most k). These modules contain nice submodules N contained in H k (M) such that M/N is a direct sum of uniserial modules. Here, we investigate the class "Equation missing" of QTAG-modules, containing nice submodules N⊆H k (M) such that M/N is totally projective. We also study strong ω-elongation of totally projective QTAG-modules by (ω+k)-projective QTAG-modules.

On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG 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. 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.

Generalizations of basic and large submodules of $$QTAG$$ -modules

A \(QTAG\)-module \(M\) over an associative ring \(R\) with unity is \(k\)-projective if \(H_k(M)=0\) and for a limit ordinal \(\sigma ,\) it is \(\sigma \)-projective if there exists a submodule \(N\) bounded by \(\sigma \) such that \(M/N\) is a direct sum of uniserial modules. \(M\) is totally projective if it is \(\sigma \)-projective for all limit ordinals \(\sigma .\) If \(\alpha \) denotes the class of all \(QTAG\)-modules \(M\) such that \(M/H_\beta (M)\) is totally projective for every ordinal \(\beta <\alpha ,\) then these modules are called \(\alpha \)-modules. Here we study these \(\alpha \)-modules and generalize the concept of basic submodules as \(\alpha \)-basic submodules. It is found that every \(\alpha \)-module \(M\) contains an \(\alpha \)-basic submodule and any two \(\alpha \)-basic submodules of \(M\) are isomorphic. A submodule \(L\) of an \(\alpha \)-module is \(\alpha \)-large if \(M=L+B,\) for any \(\alpha \)-basic submodule \(B\) of \(M.\) Many other interesting properties of \(\alpha \)-basic, \(\alpha \)-large and \(\alpha \)-modules are studied.