H-supplemented modules with respect to images of fully invariant submodules

Let R be a commutative ring with unity. Let W be an R-module, for K≤F, where F is a submodule of W and K is said to be R-annihilator coessential submodule of F in W (briefly R-a-coessential) if (denoted by K F in W). An R-module W is called strongly hollow -R-annihilator -lifting module (briefly, strongly hollow-R-a-lifting), if for every submodule F of W with hollow, there exists a fully invariant direct summand K of W such that K F in W. An R - module W is called strongly R - annihilator - ( hollow - lifting ) module ( briefly strongly R - a - ( hollow - lifting ) module ), if for every submodule F of W with R - a - hollow, there exists a fully invariant direct summand K of W such that K F in W.

Harada (cf. 1983) introduced the lifting property for maximal submodules and the extending property for simple submodules, and Oshiro (cf. 1983b) definitely introduced lifting modules and extending modules. Since then, the method and results in their works urged not only the study on (quasi-)discrete and (quasi-)continuous modules but also the one on themselves. The reader is referred to Mohamed and Müller (1990) and Dung et al. (1994) for the research on lifting modules and extending modules. In this article, we study the following fundamental open problems: When is a direct sum ⨁IMi of lifting (extending) modules {Mi}I lifting (extending)? These problems are unsolved even in the case that the index set I is finite. Now, the purpose of this article is to study the problem concerned with finite direct sums of lifting modules.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

This thesis contains two major parts. In the first part, I introduce a new theory of modules with iterative connection. This theory unifies the theory of modules with connection in characteristic zero as given by N. Katz and the theory of iterative differential modules in positive characteristic as given by B. H. Matzat und M. van der Put. The second part of this work is about the differential Abhyankar conjecture for iterative Picard-Vessiot extensions (IPV-extensions). This conjecture is concerned with the problem which linear algebraic groups occur as iterative differential Galois groups of IPV-extensions with restricted singular locus. In this thesis, I prove the differential Abhyankar conjecture for connected groups and give necessary and sufficient conditions for connected groups for being realisable with given singular points.

The question of whether all projective modules (or all finitely generated projective modules) over a given ring are free or extended is solved positively only for relatively small class of rings. In 1976, a significant progress was achieved in this area of research: D. Quillen and A. A. Suslin simultaneously and independently of one another proved that all projective modules over algebras of polynomials over fields are free modules, which was a solution to the well-known J.-P. Serre’s problem. Later similar results were obtained by various researchers for projectivemodules over ringsmore general then algebras of polynomials over fields. For example, V. A. Artamonov solved this problem for rings of quantum polynomials. The current state of this line of research is described in detail in the survey paper [1] containing an extensive list of references. Inmany cases, it is possible to prove that all projectivemodules of sufficiently large ranks over a given ring are free (or extended), but for projectivemodules of small ranks (for example, of rank one) this result does not hold. In this paper, we discuss problems of more special character: The problem of finding in an explicit form isomorphisms between a given projective module and a free (or extended) module, and, for an explicitly given projective module, the problem of efficient determination whether it is a free (or extended) module. By definition, a finitely generated projective module is a direct summand of a free module. Therefore, the most general way to define such a module is to indicate the idempotent matrix corresponding to the projection onto the direct summand. We show that, for modules over graded rings, the fact that a module is extended can be established if the sequence of powers of nonzero homogeneous components of the given idempotent matrix is known (Theorem 1), or under condition that certain blocks of positive homogeneous components of the same matrix vanish (Theorem 2). In the above described cases, there is a method of explicit constructing of the desired isomorphism. Statements of such kind are derived as consequences of the results of [2], where the author considered a situation which is so general that one can take as an “algebra” for which the results of [2] are valid not only linear multioperator algebras but modules over a certain class of rings. In this paper, we use the basic constructions, definitions, and notation from [2]. We make only insignificant changes in notation in order that it correspond to the generally adopted notation in the theory of rings and modules. So, the notation f1 used in [2] (for a polynomial, it corresponds to the homogeneous component of the first degree) is changed here by f0 since it corresponds to the zero component of the grading of the ring. Correspondingly, the other indices are also shifted by one.

The present work gives some characterizations of R-modules with the direct summand sum property (in short DSSP), that is of those R-modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct summand, too. General results and results concerning certain classes of R-modules (injective or projective) with this property, over several rings, are presented.

Keskin and Tribak (2005) studied the structure of weak lifting modules with small radicals over commutative noetherian (local) rings. They proved that a module M is weak lifting if and only if M is a direct sum of local modules of some special type. Such modules were further studied by Tribak (2007). In this note we study weak lifting modules M with small radical over arbitrary rings. We prove that M is an irredundant sum M=Σ i∈I M i where each M i is local and Σ i∈F M i is a summand of M for every finite subset F of I. Moreover Σ i∈F M i = ⊕ i∈F K i with K i local. In particular a finitely generated weak lifting module is a direct sum of local modules. This generalizes the analogous result for finitely generated lifting modules.

A module M is said to satisfy the C 12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C 11 (and hence also C 1) condition implies the C 12 condition. We show that the class of C 12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C 12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.

The objective of our research paper is to introduce as well as to study many essential properties of the concept of extending semimodules. A semimodule S is named extending (CS) if every subsemimodule of S is essential in a direct summand of S. Therefore, extending semimodule behaviour with respect to direct sums and direct summands are examined. Moreover, studying some properties of these semimodules concepts, e.g., every direct summand of a CS-semimodule is a CS-semimodule. While the direct sum of extending semimodules is not necessarily extending.

Sufficient conditions are given in this note to determine when an isomorphism between endomorphism algebras of torsion-free modules over a valuation domain induces an isomorphism between these modules. The main results are contained in Theorems 7 and 10 and are as follows: If M and N are separable, torsion-free R-modules over a valuation domain R such that M is homogeneous of type I and N homogeneous of type J with End R M ~ End R N, then, there is a homogeneous separable torsion-free R-module U such that M ~ U | I and N ~ U | J. In particular, if I ~ J, then M ~ N. Throughout the paper R denotes a communatite valuation domain with unit and Q its field of quotients; all modules will be unital torsion-free R-modules. Some of the stated results are true in a more general setting (such as modules over any domains etc.) which may be checked by examining the proofs. For terminology and supporting results on modules over valuation domains we will rely on [1]. Recall that valuation domains are characterized by the property that, for every q e Q either q ~ R or 1/q ~ R. Every rank one torsion-free R-module is isomorphic either to Q or to an ideal of R and our results that involve ideals of R will often include the case of Q as well, without a special notice. A torsion-free module is separable if every finite subset of its elements is contained in a finite rank completely decomposable direct summand or, equivalently, if the purification (a). of each of its elements a is a direct summand ([1], Lemma XIV 2.6). Our results are modeled after the corresponding results for abelian groups to be found

This chapter is the beginning of an examination of algebras that are more general than the semisimple algebras. Our strategy is to generalize the process that led to the Wedderburn Structure Theorem. The appropriate substitutes for the basic building blocks of that theory—simple modules—are indecomposable modules. These modules are introduced in this chapter. The analogue of Schur’s lemma provides a characterization of indecomposable modules in terms of their endomorphism algebras. The main result of the chapter is the Krull-Schmidt Theorem. It leads to the conclusion that finitely generated modules over an Artinian algebra decompose uniquely into direct sums of indecomposable modules. In short, the results of Chapter 2 have close analogues in the theory of modules over Artinian algebras.

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

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring $k[x]$, where $k$ denotes a field. There is some overlap with recent work of Höppner and Lenzing. The results obtained include identification of the free, projective, and injective modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. Pure submodules are direct summands if they are bounded below. In such case, the pure submodule is itself a direct sum of cyclic submodules. It is also noted that Cohen and Gluck’s Stacked Bases Theorem remains true if the modules are graded.

This work aims to introduce and explore the concept of SS-semimodules over semirings to generalize fully stable semimodules and duo semimodules. We call an S-semimodule C is SS-semimodule if every direct summand of C is stable. Equivalent expressions, properties, and related results of SS-semimodules are given. We prove that an S-semimodule C becomes SS-semimodule in case every decomposition C = H ⊕ K, HomS (H, K) = 0. Results regarding fully stable semimodules were generalized to SS-semimodules. For example, if C is a regular SS-semimodule, then EndS (C) is fully stable. Known semimodules related to SS-semimodules were investigated. We obtain that, every multiplication semimodule is SS-semimodule. Also,the SS-semimodule property satisfies the summand intersection property. Moreover, we discuss the direct sum property and inherited property for SS-semimodules.

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.