ON ALMOST PRIME SUBMODULES

In this article, we define almost prime submodules as a new generalization of prime and weakly prime submodules of unitary modules over a commutative ring with identity. We study some basic properties of almost prime submodules and give some characterizations of them, especially for （finitely generated faithful） multiplication modules.

INTERSECTION OF PRIME SUBMODULES AND DIMENSION OF MODULES

The aim of this paper is to study the conditions by which a P-prime sub-module can be expressed as a finite intersection or union of P-prime submodules. Also corresponding to dimension and rank of modules, some equivalent conditions for a ring to be a Dedekind domain are given.

The Primary Radical of a Submodule

In this paper we introduced a definition for the primary radical of a submodule with some of its basic properties. We also define the P-radical submodule and review some results about it. We find a method to characterize the primary radical of a finitely generated submodule of a free module.

Hilbert-Schmidtness of Submodules in H^(2)(D^(2))Containing(z)-β(w)

A closed subspace M of the Hardy space H^(2)(D^(2))over the bidisk is called submodule if it is invariant under multiplication by coordinate functions z and w.Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem.This paper proves that every finitely generated submodule M containing(z)-Φ(w)is Hilbert-Schmidt,where 0(z),p(w)are two finite Blaschke products.

Zariski Topologies for Coprime and Second Submodules

Let M be a non-zero module over an associative （not necessarily commutative） ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Spec-s（M） of second submodules of M and the spectrum Spec-c（M） of coprime submodules of M, study several properties of these spaces and investigate their internlay with the algebraic properties of M.

Interval Valued Intuitionistic(S,T)-fuzzy H_ν-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.

Overview on Submodule Topologies,Modeling,Modulation,Control Schemes,Fault Diagnosis,and Tolerant Control Strategies of Modular Multilevel Converters

In the present scenario,modular multilevel converters(MMCs)are considered to be one of the most promising and effective topologies in the family of high-power converters because of their modular design and good scalability;MMCs are extensively used in high-voltage and high-power applications.Based on their unique advantages,MMCs have attracted increasing attention from academic circles over the past years.Several studies have focused on different aspects of MMCs,including submodule topologies,modeling schemes,modulation strategies,control schemes for voltage balancing and circulating currents,fault diagnoses,and fault-tolerant control strategies.To summarize the current research status of MMCs,all the aforementioned research issues with representative research approaches,results and characteristics are systematically overviewed.In the final section,the current research status of MMCs and their future trends are emphasized.

Operational Reliability Model of Hybrid MMC Considering Multiple Time Scales and Multi-state Submodule

Environmental and electrical factors such as wind speed,air temperature and switching frequency have significant influences on the operational reliability of hybrid modular multilevel converter(MMC),which is commonly used for the wind power transmission.However,the existing reliability model of hybrid MMC based on statistics cannot accurately reflect the impact of these factors.In this paper,a new operational reliability model of hybrid MMC is presented.The reliability index of the hybrid MMC is coupled with its operation characteristics by calculating multi-term thermal cycling.In addition,an operation strategy of hybrid MMC is proposed to improve its reliability.The multi-state submodule(SM)is developed,which is capable of bypassing specific faulty power modules instead of the whole SM.Case studies show that the proposed operational reliability model could describe the impact of environmental and electrical factors.Also,the proposed operation strategy can improve the reliability of hybrid IMMC by extending the operation time of SMs.

The Existence of Maximal w^＊-closed Submodules in Nest Algebra Modules

Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.

Modules with Finitely Many Submodules

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.

M_(φ)-type Submodules over the Bidisk

Let H^(2)(D^(2))be the Hardy space over the bidisk D^(2),and let M_(φ)=[(z-φ(w))^(2)]be the submodule generated by(z-φ(w))^(2),whereφ(w)is a function in H∞(w).The related quotient module is denoted by N_(φ)=H^(2)(D^(2))ΘM_(φ).In the present paper,we study the Fredholmness of compression operators S_(z),S_(w) on N_(φ).Whenφ(w)is a nonconstant inner function,we prove that the Beurling type theorem holds for the fringe operator F_(w) on[(z-w)^(2)]Θz[(z-w)^(2)]and the Beurling type theorem holds for the fringe operator Fz on M_(φ)ΘwM_(φ)ifφ(0)=0.Lastly,we study some properties of F_(w) on[(z-w^(2))^(2)]Θz[(z-w^(2))^(2)].

Classification of Hardy Submodules and Characteristic Space Theory（A Brief and Selective Survey）

This paper is a brief and selective survey on classification of Hardy Submodules over the polydisk. The survey reports some progress in classification of Hardy submodules

The φ-Group for n Submodules Systems as a Generalization of the K-Theory

In this paper,the authors construct a φ-group for n submodules,which generalizes the classical K-theory and gives more information than the classical ones.This theory is related to the classification theory for indecomposable systems of n subspaces.

A Prime Submodule Principle

In this paper, we extend the concept of Ako and Ol~ families to submodules, study the behavior of the extended prime submodule principle and use these concepts to give new proofs of some familiar theorems.

Prime Submodules and Flat Modules

In this paper, some characterizations of prime submodules in flat modules and, particularly, in free modules are given. Furthermore, the height of prime submodules and some saturated chain of prime submodules are also given.

Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a^2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c > 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p > d.

Group Action on Fuzzy Modules

In this article, we introduce the notion of fuzzy G-module by defining the group action of G on a fuzzy set of a Z-module M. We establish the cases in which fuzzy submodules also become fuzzy G-submodules. Notions of a fuzzy prime submodule, fuzzy prime G-submodule, fuzzy semi prime submodule, fuzzy G-semi prime submodule, G-invariant fuzzy submodule and G-invariant fuzzy prime submodule of M are introduced and their properties are described. The homomorphic image and pre-image of fuzzy G-submodules, G-invariant fuzzy submodules and G-invariant fuzzy prime submodules of M are also established.

Notes on pure projective modules

Let R be an associated ring with identity. A new equivalent characterization of pure projective left R-modules is given by applying homological methods. It is proved that a left R-module P is pure projective if and only if for any pure epimorphism E→M→0, where E is pure injective, HomR（P, E）→HomR（P, M）→0 is exact. Also, we obtain a dual result of pure injective left R-modules. Furthermore, it is shown that every pure projective left R-module is closed under pure submodule if and only if every pure injective left R-module is closed under pure epimorphic image.

On closed weak supplemented modules

A module M is called closed weak supplemented if for any closed submodule N of M, there is a submodule K of M such that M=K+N and K∩N<<M. Any direct summand of closed weak supplemented module is also closed weak supplemented. Any nonsingular image of closed weak supplemented module is closed weak supplemented. Nonsingular V-rings in which all nonsingular modules are closed weak supplemented are characterized in Section 4.

Relatively Injective Modules with Respect to Torsion Theory

For a hereditary torsion theory τ, this paper mainly discuss properties of A-τ-injective modules, where A is a fixed left R-module. It is proved that if M is an A-τ-injective, B is a submodule of A, then 1） M is A/B-τ-injective; 2） M is B-τ-injective when B is τ-dense in A. Furthermore, we show that if A1,A2,... An, are relatively injective modules, then A1 A2 ... An is self-τ-injective if and only if A1 is self-τ-injective for each i.