Modules Whose Submodules are Essentially Embedded in Direct Summands

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).

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.

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.

Sublattices of the Lattice of Pre-natural Classes of Modules

In this paper we introduce the concept of im-summand coinvariance and im-small coinvariance; that is, a module $M$ over a right perfect ring is said to be im-summand (im-small) coinvariant if, for any endomorphism $\varphi$ of $P$ such that ${\rm Im} \varphi$ is a direct summand (a small submodule) of $P$, $\varphi (\ker \nu) \subseteq \ker \nu$, where $(P, \nu)$ is the projective cover of $M$. We first give some fundamental properties of im-summand coinvariant modules and im-small coinvariant modules, and we prove that, for modules $M$ and $N$ over a right perfect ring such that $N$ is a small epimorphic image of $M$, $M$ is $N$-im-summand coinvariant if and only if $M$ is (im-coclosed) $N$-projective. Moreover, we introduce ker-summand invariance and ker-essential invariance as the dual concept of im-summand coinvariance and im-small coinvariance, respectively, and show that, for modules $M$ and $N$ such that $N$ is isomorphic to an essential submodule of $M$, $M$ is $N$-ker-summand invariant if and only if $M$ is (ker-closed) $N$-injective.

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

Lifting modules plays important roles in module theory. H-supplemented modules are a nice generalization of lifting modules which have been studied extensively recently. In this article, we introduce a proper generalization of H-supplemented modules via images of fully invariant submodules. Let F be a fully invariant submodule of a right Rmodule M. We say that M is IF -H-supplemented in case for every endomorphism φ of M, there is a direct summand D of M such that φ(F) + X = M if and only if D + X = M, for every submodule X of M. It is proved that M is IF -H-supplemented if and only if F is a dual Rickart direct summand of M for a fully invariant noncosingular submodule F of M. It is shown that the direct sum of IF –H supplemented modules is not in general IF -H-supplemented. Some sufficient conditions such that the direct sum of IF -H-supplemented modules is IF -H-supplemented are given

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.

Grothendieck groups of subcategories

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

The polynomial ring over a Goldie ring need not be a Goldie ring

We study the structure of infinitely generated small modules over abelian regular rings, i.e., modules over which the covariant functor Hom commutes with direct sums. It is shown that every infinitely generated small module has either an infinitely generated factor which is at most 22ω -generated or a countably generated essential submodule. As a consequence, we prove a module-theoretic criterion of steadiness for abelian regular rings.

Direct-Sum Decompositions over Local Rings