Results on weakly uniserial modules

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

A uniserial module is a module that satisfies both ascending chain condition and descending chain condition, which makes a uniserial module an Artinian module and a Noetherian module at the same time. Recently an algebraic structure from ring theory, called almost prime ideal, is generalized into a module theory and called an almost prime submodule. Some researchers have examined the characterizations of this new algebraic structure in various types of modules. In this article, we provide some insights into the almost prime submodule of a uniserial cyclic module In this study, we have discovered that the non-zero almost prime submodule of the cyclic uniserial module is unique.

The purpose of this thesis is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being -torsion and -torsionfree modules as well as -dense and -pure submodules. In the rst chapter we also introduce a new concept, that of a -compact module, which is basic enough to deserve a place among the preliminaries. The results that we obtain fall into three areas which are to a certain degree interrelated. The rst area is on -Max modules, which we introduce as a torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsiontheoretic counterparts which we utilize in our generalization. Furthermore, we de ne and characterize left -Max rings and apply the torsion-theoretic version of Shock's theorem to obtain a characterization of -short modules motivated by a recent article in which short modules were introduced. The second area deals with various avours of -injectivity, some known and some new. We introduce -M -injective and s-M -injective modules and examine their relationship with the known concepts of -injective and -quasiinjective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-M -injective modules, and give a generalization of Azumaya's Lemma. We also prove that every M -generated module has a -M -injective hull which is unique up to isomorphism and show how this is linked to the -quasi-injective hull. We then examine Σ-injectivity, generalizing well-known results by Faith, Albu and N ast asescu and Cailleau which provide necessary and su cient conditions for the Σinjective property, the Σ-s-M -injective property and for a direct sum of Σ-s-M -injective modules to be Σ-s-M -injective. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. The approach is twofold. The rst is via -CS modules which serve as a generalization of CS modules as well as -quasi-continuous, -quasi-injective and -injective modules, and the second is via s-CS modules which are a special case of CS modules. Our motivation is to provide a torsion-theoretic analogue of a well-known result by Okado which characterizes Noetherian modules. We have some partial results using s-CS modules and a nice torsion-theoretic analogue, albeit without the use of -CS or s-CS modules. We also examine the relationship between our relative versions of CS modules with those of other authors and obtain re nements to some of their results.

This paper contains a review of results on endomorphism rings of Abelian groups. On one hand, this rapidly developing section of contemporary algebra can be considered as a part of Abelian group theory; on the other hand, it can be considered as a branch of the theory of endomorphism rings of modules. This section is close to both theories, but it has many specific features. There are several important reasons to study endomorphism rings of Abelian groups. First, it provides us with new information on these groups. Second, it stimulates the study of the theory of modules and their endomorphism rings. There are other fields of algebra where the application of endomorphism rings can be useful (additive groups of rings, E-modules and E-rings, and so on). There are already many excellent results in the theory of endomorphism rings of Abelian groups. A large number of methods are used (for example, group, module, categorical, topological, and set-theoretical methods). This survey gives a satisfactory overview of the content and methods of this part of mathematics. One chapter of the book of L. Fuchs [128] is devoted to endomorphism rings of Abelian groups. Also, they are considered in the works of I. Kaplansky [183], A. G. Kurosh [216], D. Arnold [29], and K. Benabdallah [51]. A number of results in this field of algebra are considered in the surveys of A. P. Mishina [244–248], A. V. Mikhalev [242], A. V. Mikhalev and A. P. Mishina [243], V. T. Markov, A. V. Mikhalev, L. A. Skornyakov, and A. A. Tuganbaev [233]. The work of R. Baer [44] has played an important role in the making of the theory of endomorphism rings of Abelian groups and modules. Several fields of ring theory related to endomorphism rings of modules are considered in the works of I. Lambek [218], C. Faith [109, 110], F. Kasch [184], A. A. Tuganbaev [328, 329], and others. However, there does not exist a book which is especially devoted to endomorphism rings; also, there does not exist a systematical presentation of the main results of this theory. This survey slightly fills in this appreciable gap. We note that the book of L. Fuchs does not reflect all the fields of the theory of endomorphism rings. In addition, new sections of this theory appeared after the publication of this book; also, several excellent results were obtained in traditional sections of the theory. In this paper, we consider the main fields of the theory of endomorphism rings of Abelian groups. The most typical results of this theory are included in the review. Some theorems are restated (some statements are presented in more general form, and some theorems are presented in a simplified form). In Secs. 8 and 9, we use the papers of May [235], Gobel [136], and Corner–Gobel [74]. Some parts of these papers are included in the corresponding sections. Our review is designed for specialists in the theory of Abelian groups and the theory of rings and modules. We wish to avoid routine recounting of results and try to give an idea of the possibilities, methods of proofs, and relations between various research fields and individual results. Unsolved problems are presented at the end of every section. Most of them are known; we only systematize them. The authors do not wish to present a complete bibliography; also, there are difficulties related to precedence. Every Abelian group belongs to exactly one of the following three classes of groups: torsion groups, torsion-free groups, and mixed groups. (A group is said to be mixed if it contains a nonzero element of finite order and an element of infinite order.) The properties of endomorphism rings of groups of these three classes are often different. Usually, we emphasize this fact.

Introduction. Our aim is to apply category concepts to the study of commutative C* algebras. In particular we shall study injective hulls. This appears to have been overlooked in the literature, although injective Banach spaces have been studied extensively. However, in [2] the dual category of compact Hausdorff spaces is studied. It is shown that the projectives are precisely the extremely disconnected spaces and that projective covers always exist. Hence, injective hulls always exist in the original category. Injective hulls are very important in ring theory, where they are used to obtain generalized quotient rings. They play a critical role in the proof of the Mitchell embedding theorem for abelian categories. They lead also to a novel way of obtaining the decomposition theorem for commutative noetherian rings. Recently Isbell constructed injective hulls in the category of metric spaces [5]. Injective hulls in Banach spaces are discussed in [1] and [4].

Singular Submodule and Injective Hull

Uniserial modules over valuation rings

An effective way to understand the behavior of a ring R is to study the various ways in which R acts on its left and right modules. Thus, the theory of modules can be expected to be an essential chapter in the theory of rings. Classically, modules were used in the study of representation theory (see Chapter 3 in First Course). With the advent of homological methods in the 1950s, the theory of modules has become much broader in scope. Nowadays, this theory is often pursued as an end in itself. Quite a few books have been written on the theory of modules alone.KeywordsCommutative RingProjective ModuleFree ModuleDivision RingInjective ModuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The themes of cancellation, internal cancellation, substitution and exchange have led to a lot of interesting research in the theory of modules over commutative and noncommutative rings. This article provides a quick and relatively self-contained introduction to the voluminous work in this area, using the notion of the stable range of rings as a unifying tool. With only a small number of exceptions, all theorems stated here are proved in full, modulo basic facts in the theory of modules and rings available in standard textbooks on ring theory.

In this survey paper we illustrate a general strategy which consists on putting a module-theoretical result into a latticial frame (we call it latticization), in order to translate that result to Grothendieck categories (we call it absolutization) and module categories equipped with hereditary torsion theories (we call it relativization). The renowned Hopkins–Levitzki Theorem and Osofsky–Smith Theorem from Ring and Module Theory, we will discuss in the last two sections of the paper, are among the most relevant illustrations of the power of this strategy.

Quadratic maps between modules

This paper introduces the concept of graded Jgr-2-absorbing primary submodules, a new intermediate structure in graded module theory. Motivated by the need to extend and unify classical notions such as graded prime and graded primary submodules, the study develops a comprehensive framework describing their defining characteristics and relationships. Through a sequence of rigorous theorems and illustrative examples, we establish fundamental properties, equivalence conditions, and inclusion relations that clarify the behavior of these submodules within graded modules. The findings show that graded Jgr-2-absorbing primary submodules generalize several known structures while maintaining distinctive algebraic flexibility. Moreover, the results concerning graded homomorphisms and multiplication modules demonstrate the robustness of the concept and its potential applications in graded algebra. Overall, this work deepens the theoretical understanding of graded algebraic systems and provides a foundation for further research in module and ring theory.

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). Our results in this direction complement those of Iovanov-Mesyan-Reyes. An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equivalences and implications between which we prove. In particular, all the conditions are equivalent for topological rings with a countable base of neighborhoods of zero and for topologically right coherent topological rings. Considering the rings of endomorphisms of modules as topological rings with the finite topology, we establish a close connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. We show that any topologically agreeable split abelian category is Grothendieck and semisimple. We also prove that a module $\Sigma$-coperfect over its endomorphism ring has a perfect decomposition provided that either the endomorphism ring is commutative or the module is countably generated, partially answering a question of Angeleri Hugel and Saorin.

In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.

In this paper we derive some of the structure of semiperfect FPF rings. A ring is right FPF if every f.g. faithful right module is a generator. For semiperfect right and left FPF rings we show that if all one sided zero divisors are two sided zero divisors, then the classical and maximal quotient rings coincide (all four of them) and are self-injective. We show that if the intersection of the powers of the Jacobson radical is zero, then right and left regular elements are regular. Also, we show right FPF semiperfect rings contain the singular submodule of their injective hulls and that every finitely generated module contained in the injective hull and containing the ring is isomorphic to the ring.