Extending Semimodules over Semirings

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.

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.

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

Direct-Sum Decompositions over Local Rings

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R ( M / K , M ) ≠ 0 . Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

Villem Tamm, who was called also as Tamme Villu or simply Villu, was born on 10th of February 1942 in Viljandi county Kabala. He studied seven years in Villavere school and then in high school in Viljandi. He graduated Tartu State University as „economic cybernetics” (mathematical economists) in 1969. His course mates considered him as a capable and talented person and predicted him successful and fast career as a lecturer and scientist. Villu worked at the Faculty of Economics and Business Administration, University of Tartu, almost 40 years. He started as assistant and continued as senior lecturer, associate professor in the Chair of Economic Cybernetics and Statistics (during the years 1991-1992 he was the head of this chair). During the years 1993-1996 he was professor of statistics and also the head of the Institute of Informatics and Economic Modelling. In the following years he worked as associate professor of statistics and retired in 2007. The new specialisation in the soviet economic education with the name “economic cybernetics” was opened at the Tartu State University in 1967 and there was an urgent need for academic staff who could teach the subjects of specialisation. Villem Tamm was assigned to teach basic principles of “economic cybernetics” and mathematical and statistical methods. He was one of the main lecturers who were teaching the specialisation subjects of “economic cybernetics” and also one of the main supervisors of students’ research papers and Diploma Thesis. He has remarkable contribution in preparation of almost 450 students, who have graduated the specialisation of “economic cybernetics”. In essence he is one of the founders of the new direction in education of economists – “economic cybernetics” - in Estonia. During his career he was teaching different statistical and mathematical planning courses in Estonian and in Russian for stationary students and for Open University students. For those students, who specialised to economic modelling, Villu was teaching subjects like principles of economics prognosis, input-output tables, Leontief’s economic equilibrium models, system analysis in economics, etc. Lot of students from Parnu and Narva Collage, private universities of Audentes and Veritas can consider him also as their lecturer. Students evaluated him highly as a lecturer, as he was concrete in his taught subjects, he was thought-provoking and the personality to his lectures added his smugly humour. He was innovative and quickly adopted new technics and methods to modernise the teaching process. For example in 1980 he made dozens of dispositive about statistics, prepared punched card test for controlling students’ knowledge. In 2003 together with his colleague he designed a web-based course “Statistics”, which get the first price in the faculty’s competition. In total Villem Tamm has published nine textbooks and voluminous methodological materials for six courses. The way to the candidate (Ph.D) degree was not so quick. He continued his post-graduate studies in 1977 and defended Ph.D thesis (candidate in economics) in 1985. His main research areas have been very diverse: starting from the studies devoted to efficiency and effective use of labour resources, regularity of working processes up to problems and misunderstandings of statistical information. During several decades he was doing research in the field of input-output tables. From his personality Villem Tamm had good leadership abilities and as a head of different structural units he was tolerant, steady and fair leader. He had desire to motivate and direct both students and his colleagues; at the same time being dispassionate in communication and dignified in his behaviour. He was helpful and mindful of others; he had skills to solve complicated situations humanly and peacefully. During his working career he had also to fulfil different administrative obligations. He was 15 years supervisor and leader for different courses, curator of cooperation agreement between economic cybernetics from Vilnius, Latvia and Tartu Universities, member of teaching methodology commission, member of faculty’s council, etc. He was one of the founders of Estonian Statistical Society, alumni society “Hermes”, lecturer of association “Teadus” (“Science”), etc. He was participating in conducting population censuses, where he could implement his statistical knowledge and organiser’s abilities. Since the independence of Estonia he was several times leading the local election commissions during the parliament and local elections. Even if sometimes others could not see active engagement to these named and also to lot of others not mentioned activities, everything was functioning and was done at high quality.

A module M is defined to be a CS module, if every submodule of M is essential in a direct summand of M. In this paper we show, among other results, that for a right nonsingular ring R, all direct sums of CS right R-modules are CS if and only if R is a right artinian ring and every indecomposable infective right R-module has length at most two.KeywordsDirect SummandDivision RingInjective ModuleEndomorphism RingSemiprimary RingThese 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.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].