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In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism [Formula: see text], with [Formula: see text] indecomposable (uniform)... more
In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism [Formula: see text], with [Formula: see text] indecomposable (uniform) quasi-injective right [Formula: see text]-modules, and [Formula: see text] an extension of [Formula: see text] where [Formula: see text] denotes the injective hull, what is the relation between kernels of [Formula: see text] and [Formula: see text], their monogeny classes and their upper parts?
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summary:The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings... more
summary:The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three two-sided ideals, and chain rings with ``many'' two-sided ideals. We prove that there exists an $\omega_{1}$-generated uniserial module over every non-artinian nearly simple chain ring and over chain rings containing an uncountable strictly increasing (resp. decreasing) chain of right (resp. two-sided) ideals. As a consequence we describe right steady serial rings
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This paper introduces the notion of essentially ADS (e-ADS) modules. Basic structural properties and examples of e-ADS modules are presented. In particular, it is proved that (1) The class of all e-ADS modules properly contains all ADS as... more
This paper introduces the notion of essentially ADS (e-ADS) modules. Basic structural properties and examples of e-ADS modules are presented. In particular, it is proved that (1) The class of all e-ADS modules properly contains all ADS as well as automorphism invariant modules. e-ADS modules serves also as a tool for characterization of various classes of rings. It is shown that: (2) R is a QF-ring if and only if every projective right R-module is e-ADS; (3) R is a semisimple Artinian ring if and only if every e-ADS module is injective. The nal part of this paper describes properties of e-ADS rings, which allow to prove a criterion of e-ADS modules for non-singular rings: (4) Let R be a right non-singular ring and Q be its the right maximal ring of quotients. Then R is a right e-ADS ring if and only if either eQ 6 = (1 e)Q for any idempotent e 2 R or R = M2(A) for a suitable right automorphism invariant ring A.
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The aim of the paper is to describe autocompact objects in Ab5categories, i.e. objects in cocomplete abelian categories with exactness preserving filtered colimits of exact sequences, whose covariant Hom-functor commutes with copowers of... more
The aim of the paper is to describe autocompact objects in Ab5categories, i.e. objects in cocomplete abelian categories with exactness preserving filtered colimits of exact sequences, whose covariant Hom-functor commutes with copowers of the object itself. A characterization of non-autocompact object is given, a general criterion of autocompactness of an object via the structure of its endomorphism ring is presented and a criterion of autocompactness of products is proven.
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Abstract. In this paper, the categorial property of compactness of an object, i. e. commuting of the corresponding Hom functor with coproducts, is studied in categories of S-acts and corresponding structural properties of compact S-acts... more
Abstract. In this paper, the categorial property of compactness of an object, i. e. commuting of the corresponding Hom functor with coproducts, is studied in categories of S-acts and corresponding structural properties of compact S-acts are shown. In order to establish a general context and to unify approach to both of the most important categories of S-acts, the notion of a category with unique decomposition of objects is defined and studied.
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A (right R-) module M is dually slender if the covariant HomR (M, ) functor commutes with direct sums. A ring R is right steady provided that the dually slender modules coincide with the finitely generated ones. Rings satisfying various... more
A (right R-) module M is dually slender if the covariant HomR (M, ) functor commutes with direct sums. A ring R is right steady provided that the dually slender modules coincide with the finitely generated ones. Rings satisfying various finiteness conditions are known to be steady. We investigate steadiness of the rings such that each two-sided ideal is countably generated. We prove e.g. that such rings are steady provided that they are von Neumann regular and with all primitive factors artinian. Also, we obtain a characterization of steadiness for arbitrary valuation rings, and an example showing that steadiness is not left-right symmetric even for countable rings. Steady rings were introduced by Rentschler [R2] as part of an investigation of commutativity properties of covariant and contravariant Hom functors in module categories (see [W] for a survey). More recently, steady rings have played an important role in dealing with various particular problems such as investigations of h...
Research Interests: Mathematics and CRC
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Abstract The set of all elements r of a ring R such that is a unit for every unit u extends the Jacobson radical J(R). R is a UJ ring (ΔU ring, respectively) if its units are of the form ( respectively). Using a local characterization of... more
Abstract The set of all elements r of a ring R such that is a unit for every unit u extends the Jacobson radical J(R). R is a UJ ring (ΔU ring, respectively) if its units are of the form ( respectively). Using a local characterization of ΔU rings, we describe structure of group rings that are UJ rings; if RG is a UJ group ring, then R is a UJ ring, G is a 2-group and, for every nontrivial finitely generated subgroup H of G, the commutator subgroup of H is proper subgroup of H. Conversely, if R is a UJ ring and G a locally finite 2-group, then RG is a UJ ring. In particular, if G is solvable, RG is a UJ ring if and only if R is UJ and G is a 2-group.
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The paper is focused on questions when some homological and submodule-chain conditions satisfied by a module [Formula: see text] are preserved by the group module [Formula: see text]. Namely, it is proved for a group [Formula: see text]... more
The paper is focused on questions when some homological and submodule-chain conditions satisfied by a module [Formula: see text] are preserved by the group module [Formula: see text]. Namely, it is proved for a group [Formula: see text] and an [Formula: see text]-module [Formula: see text] that [Formula: see text] is flat if and only if [Formula: see text] is flat, and [Formula: see text] is artinian if and only if [Formula: see text] is artinian and [Formula: see text] is finite, which are two questions raised by Yiqiang Zhou: On Modules Over Group Rings, Noncommutative Rings and Their Applications LENS July 1-4, 2013.
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In this paper, we introduce type absolute direct summand (type-ADS) modules and rings as a natural generalization of ADS ones. Besides basic properties and characterizations of the notion, we present several examples illustrating borders... more
In this paper, we introduce type absolute direct summand (type-ADS) modules and rings as a natural generalization of ADS ones. Besides basic properties and characterizations of the notion, we present several examples illustrating borders of the theory. We also show that some particular classical classes of rings, such as commutative or right non-singular rings are type-ADS.
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A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and... more
A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.
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ABSTRACT Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content") contained in the publications on our platform. However, Taylor & Francis, our agents, and our... more
ABSTRACT Taylor & Francis makes every effort to ensure the accuracy of all the information (the "Content") contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. A right module M over a ring R is said to be retractable if Hom R NN = 0 for each nonzero submodule N of M. We show that M ⊗ R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.
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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... more
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.
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A module M is called “self-small” if the functor Hom(M, −) commutes with direct sums of copies of M. The main goal of the present article is to construct a non-self-small product of self-small modules without nonzero homomorphisms between... more
A module M is called “self-small” if the functor Hom(M, −) commutes with direct sums of copies of M. The main goal of the present article is to construct a non-self-small product of self-small modules without nonzero homomorphisms between distinct ones and to correct an error in a claim about products of self-small modules published by Arnold and Murley in a fundamental article on this topic. The second part of the article is devoted to the study of endomorphism rings of self-small modules.