Complement Submodules Research Articles

A Research on the Generalizations of Modules Whose Submodules are Isomorphic to a Direct Summand

A module M is called virtually semisimple (resp. virtually extending) if every submodule (resp. complement submodule) of M is isomorphic to a direct summand of M. It is known that virtually extending modules is a generalization of virtually semisimple modules. In this paper, the relationships between virtually extending modules and other generalizations of virtually semisimple modules are examined. Moreover, we introduce a new generalization of virtually semisimple modules; namely CH modules: We say a module M is a c-epi-retractable (or briefly CH module) if any complement submodule of M is a homomorphic image of M. CH modules contains the class of virtually extending modules and the class of epi-retractable modules. We also give some basic properties of this new module class.

On modules and rings in which complements are isomorphic to direct summands

A right R-module M is virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if is a VC2-module where denotes the injective hull, then M is injective. Communicated by Scott Chapman

On NCS modules and rings

We give some conditions as to when a (finite) direct sum of NCS modules (if no non-zero complement submodule is small) is again NCS, and also provide various conditions to be sufficient for a submodule of an NCS module to be NCS. Some comparisons between QF-rings, GP-injective rings, AGP-rings and NCS rings are investigated.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

On C-co-epi-retractable Modules

In this paper, we introduce the notion of c-co-epi-retractable modules. An R-module M is called c-co-epi-retractable if it contains a copy of its factor module by a complement submodule. The ring R is called c-co-pri if RR is c-co-epi-retractable. Conditions are found under which, a c-coepi-retractable module is extending, retractable, semi-simple, quasi-injective, injective and simple. Also, we investigate when c-co-epi-retractable modules have finite uniform dimension. Finally, right SI-rings, semi-simple artinian rings and quasi-Frobenius rings are characterized in termes of c-co-epi-retractable modules.

Some Results on Pure Submodules Relative to Submodule

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

Some Results on Pure Submodules Relative to Submodule

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, N?AM=AN+T?(N?AM). In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending. We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M. If M has the T-PIP, then N is T-purely extending.

Notes on Approximately Pure Submodules

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

Notes on Approximately Pure Submodules

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

Strongly Large Module Extensions

Abstract In this paper, we introduce strongly large submodules and investigate their properties. A submodule N of a right R-module M is said to be strongly large in case for any m ∈ M, s ∈ R with ms ̸= 0 there exists an r ∈ R such that mr ∈ N and mrs ̸= 0. In this note, we also define and study strongly large closed submodules and strongly large complement submodules

When Is a Simple Ring Noetherian?

A moduleMis known to be a CS-module (or an extending module) if every complement submodule ofMis a direct summand. It is shown that (i) a simple ringRmust be right noetherian if every cyclic singular rightR-module is CS, and (ii) over a simple ringRif every proper cyclic right module is quasi-injective, thenRis right hereditary and right noetherian.

Pure Baer injective modules

In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure Baer injective module. Some properties and some characterization of such modules are established. We also introduce two notions closely related to pure Baer injectivity; namely, the notions of a ∑‐pure Baer injective module and that of SSBI‐ring. A ring R is an SSBI‐ring if and only if every smisimple R‐module is pure Baer injective. To investigate such algebraic structures we had to define what we call p‐essential extension modules, pure relative complement submodules, left pure hereditary rings and some other related notions. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of pure split modules.

Correspondence Theorems for Nondegenerate Modules and their Endomorphism Rings

Let R U be a left R-module whose Morita context is nondegenerate and S = End(U) . We show the following: (1) There is a projectivity (that is, an order-preserving bijection) between the complement submodules of R U and the complement left ideals of S; (2) S is a left CS ring if and only if R U is a CS module; (3) S is a Baer and left CS ring if and only if R U is a nonsingular and CS module. Special cases include some earlier works.

A Lattice Isomorphism Theorem for Nonsingular Retractable Modules

AbstractLet RM be a nonsingular module such that B = EndR(M) is left nonsingular and has as its maximal left quotient ring, where is the injective hull of RM. Then it is shown that there is a lattice isomorphism between the lattice C(M) of all complement submodules of RM and the lattice C(B) of all complement left ideals of B, and that RM is a CS module if and only if B is a left CS ring. In particular, this is the case if RM is nonsingular and retractable.

Correspondence theorems for nondegenerate modules and their endomorphism rings

Let R U _RU be a left R-module whose Morita context is nondegenerate and S = End ( U ) S = {\text {End}}(U) . We show the following: (1) There is a projectivity (that is, an order-preserving bijection) between the complement submodules of R U _RU and the complement left ideals of S; (2) S is a left CS ring if and only if R U _RU is a CS module; (3) S is a Baer and left CS ring if and only if R U _RU is a nonsingular and CS module. Special cases include some earlier works.

Cyclic modules whose quotients have all complement submodules direct summands

Cyclic modules whose quotients have all complement submodules direct summands

Nonsingular retractable modules and their endomorphism rings

A module RM is said to be retractable if HomR (M, U) ≠ 0 for each nonzero submodule U of M. M is said to be a CS module if every complement submodule of M is a direct summand in M. Retractable modules are compared to nondegenerate modules on the one hand and to e–retractable modules on the other (nondegenerate implies retractable implies e–retractable); and it is shown that if M is nonsingular and retractable, then EndRM is a left CS ring if and only if M is a CS module.