Infinite Direct Sums of Lifting Modules

A module M over a ring R is called a lifting module if every submodule A of M contains a direct summand K of M such that A/K is a small submodule of M/K (e.g., local modules are lifting). It is known that a (finite) direct sum of lifting modules need not be lifting. We prove that R is right Noetherian and indecomposable injective right R-modules are hollow if and only if every injective right R-module is a direct sum of lifting modules. We also discuss the case when an infinite direct sum of finitely generated modules containing its radical as a small submodule is lifting.     

Lifting Modules with Indecomposable Decompositions

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.     

On strongly ⊕-supplemented modules

Strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are defined and some properties of strongly ⊕-supplemented and strongly cofinitely ⊕-supplemented modules are investigated. Let R be a ring. Then every R-module is strongly ⊕-supplemented if and only if R is perfect. The finite direct sum of ⊕-supplemented modules is ⊕-supplemented. However, this is not true for strongly ⊕-supplemented modules. Any direct sum of cofinitely ⊕-supplemented modules is cofinitely ⊕-supplemented but this is not true for strongly cofinitely ⊕-supplemented modules. We also prove that a supplemented module is strongly ⊕-supplemented if and only if every supplement submodule lies above a direct summand.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

Lifting Modules Over Right Perfect Rings

A module M is called extending if, for any submodule X of M, there exists a direct summand of M which contains X as an essential submodule, that is, for any submodule X of M, there exists a closure of X in M which is a direct summand of M. Dually, a module M is said to be lifting if, for any submodule X of M, there exists a direct summand of M which is a co-essential submodule of X, that is, for any submodule X of M, there exists a co-closure of X in M which is a direct summand of M.

Okado (1984 Okado , M. ( 1984 ). On the decomposition of extending modules . Math. Japonica 29 : 939 – 941 . [Google Scholar]) has studied the decomposition of extending modules over right noetherian rings. He obtained the following: A ring R is right noetherian if and only if every extending R-module can be expressed as a direct sum of indecomposable (uniform) modules.

In this article, we show that every (finitely generated) lifting module over a right perfect (semiperfect) ring can be expressed as a direct sum of indecomposable modules. And we consider some application of this result.     

Modules in Which Every Fully Invariant Submodule is Essential in a Direct Summand

A module M is called extending if every submodule of M is essential in a direct summand. We call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the general module setting. For example, in contrast to extending modules, a direct sum of FI-extending modules is FI-extending. Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself). Again, unlike the extending property, the FI-extending property is shown to carry over to matrix rings. Several results on ring direct decompositions of FI-extending rings are obtained, including a proper generalization of a result of C. Faith on the splitting-off of the maximal regular ideal in a continuous ring.     

Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

A widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result.     

Properties of <Emphasis Type="Italic">P</Emphasis>-coherent and Baer modules

M is called a P-coherent (resp. PP) module if its every principal submodule is finitely presented (resp. projective). M is said to be a Baer module if the annihilator of its every subset is a direct summand of R. In this paper, we investigate the properties of P-coherent, PP and Baer modules. Some known results are extended.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

MODULES WITH FULLY INVARIANT SUBMODULES ESSENTIAL IN FULLY INVARIANT SUMMANDS

ABSTRACT

A module M is called (strongly) FI-extending if every fully invariant submodule is essential in a (fully invariant) direct summand. The class of strongly FI-extending modules is properly contained in the class of FI-extending modules and includes all nonsingular FI-extending (hence nonsingular extending) modules and all semiprime FI-exten ding rings. In this paper we examine the behavior of the class of strongly FI-extending modules with respect to the preservation of this property in submodules, direct summands, direct sums, and endomorphism rings.     

On generalization of ⊕-cofinitely supplemented modules

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, cgs ^⊕-modules. It is shown that a module with summand sum property (SSP) is cgs ^⊕ if and only if M/w Loc^⊕ M (w Loc^⊕ M is the sum of all w-local direct summands of a module M) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending cgs ^⊕-module is cgs ^⊕, and that, for any ring R, every free R-module is cgs ^⊕ if and only if R is semiperfect.     

t-Extending Modules and t-Baer Modules

We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z ₂(R _R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective.     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.     

Strongly radical supplemented modules

Z?schinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a nonlocal Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

Singular and nonsingular modules relative to a torsion theory

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.