scholarly journals A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

2014 ◽  
Vol 14 (02) ◽  
pp. 1550016 ◽  
Author(s):  
N. R. Baeth ◽  
A. Geroldinger ◽  
D. J. Grynkiewicz ◽  
D. Smertnig
Keyword(s):  
Combinatorial Problems ◽  
Point Of View ◽  
Unique Element ◽  
Theoretical Point ◽  
Finitely Generated ◽  
Prime Rings ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Krull Monoid

Let R be a ring and let [Formula: see text] be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let [Formula: see text] denote a set of representatives of isomorphism classes in [Formula: see text] and, for any module M in [Formula: see text], let [M] denote the unique element in [Formula: see text] isomorphic to M. Then [Formula: see text] is a reduced commutative semigroup with operation defined by [M] + [N] = [M ⊕ N], and this semigroup carries all information about direct-sum decompositions of modules in [Formula: see text]. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R(M) is semilocal for all [Formula: see text], then [Formula: see text] is a Krull monoid. Suppose that the monoid [Formula: see text] is Krull with a finitely generated class group (for example, when [Formula: see text] is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case, we study the arithmetic of [Formula: see text] using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid [Formula: see text] for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free 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).

scholarly journals Decomposition of Games: Some Strategic Considerations

2021 ◽  
Author(s):  
Joseph Abdou ◽  
Nikolaos Pnevmatikos ◽  
Marco Scarsini ◽  
Xavier Venel
Keyword(s):  
Point Of View ◽  
Theoretical Point ◽  
Direct Sum ◽  
Affine Mappings ◽  
Finite Games ◽  
Potential Harmonic

Orthogonal direct-sum decompositions of finite games into potential, harmonic and nonstrategic components exist in the literature. In this paper we study the issue of decomposing games that are strategically equivalent from a game-theoretical point of view, for instance games obtained via transformations such as duplications of strategies or positive affine mappings of the payoffs. We show the need to define classes of decompositions to achieve commutativity of game transformations and decompositions.

On a class of separable modules

2021 ◽  
pp. 2250034
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Finitely Generated ◽  
Direct Sums ◽  
Simple Modules ◽  
Direct Sum ◽  
The Stability

A module [Formula: see text] is called [Formula: see text]-separable if every proper finitely generated submodule of [Formula: see text] is contained in a proper finitely generated direct summand of [Formula: see text]. Indecomposable [Formula: see text]-separable modules are shown to be exactly the simple modules. While direct summands of an [Formula: see text]-separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of [Formula: see text]-separable modules is [Formula: see text]-separable. Also, we prove that if [Formula: see text] and [Formula: see text] are [Formula: see text]-separable modules such that [Formula: see text] is [Formula: see text]-projective, then [Formula: see text] is [Formula: see text]-separable. We conclude the paper by providing some characterizations of several classes of rings in terms of [Formula: see text]-separable modules. Among others, we prove that the class of rings [Formula: see text] for which every (injective) [Formula: see text]-module is [Formula: see text]-separable is exactly that of semisimple rings.

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

1987 ◽  
Vol 39 (2) ◽  
pp. 365-427 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Positive Cone ◽  
Point Of View ◽  
Order Structure ◽  
Topological Spaces ◽  
Projective Modules ◽  
Finitely Generated ◽  
C Algebras ◽  
Isomorphism Classes

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

scholarly journals The identity of certain representation algebra decompositions

1970 ◽  
Vol 3 (1) ◽  
pp. 73-74
Author(s):  
S. B. Conlon ◽  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Linear Algebra ◽  
Residue Field ◽  
Complex Field ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
A Finite Group ◽  
Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

Modules Behaving Like Torsion Abelian Groups

1979 ◽  
Vol 22 (4) ◽  
pp. 449-457 ◽  
Author(s):  
M. Zubair Khan
Keyword(s):  
Homomorphic Image ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Prime Rings ◽  
Direct Sum ◽  
Uniserial Modules

Recently H. Marubayashi [1,2] and S. Singh [10,11,12] generalized some results of torsion abelian groups for modules over some restricted rings, like bounded Dedekind prime rings, bounded hereditary Noetherian prime rings. Singh [12] introduced the concept of h-purity for a module MR satisfying the following conditions:(I) Every finitely generated submodule of every homomorphic image of M is a direct sum of uniserial modules.

Timber Auctions in the Context of the Modern Auction Theory

2007 ◽  
pp. 86-94
Author(s):  
A. Manakov
Keyword(s):  
Theoretical Analysis ◽  
Auction Theory ◽  
Point Of View ◽  
Theoretical Point ◽  
English Auction ◽  
Analysis And Evaluation

The article provides theoretical analysis and evaluation of the timber auctions reforms in Russia. The author shows that the mechanism of the "combined auctions", which functioned until recently, is more appropriate from the theoretical point of view (and from the point of view of the Russian practice) as compared to the officially approved format of the English auction.

A novel cryptosystem based on abstract automata and Latin cubes

2015 ◽  
Vol 52 (2) ◽  
pp. 221-232
Author(s):  
Pál Dömösi ◽  
Géza Horváth
Keyword(s):  
Block Cipher ◽  
Finite Automata ◽  
Theoretical Background ◽  
Point Of View ◽  
Theoretical Point ◽  
Transition Matrices ◽  
Encryption And Decryption ◽  
Secret Keys ◽  
Automata Network ◽  
Fully Functioning

In this paper we introduce a novel block cipher based on the composition of abstract finite automata and Latin cubes. For information encryption and decryption the apparatus uses the same secret keys, which consist of key-automata based on composition of abstract finite automata such that the transition matrices of the component automata form Latin cubes. The aim of the paper is to show the essence of our algorithms not only for specialists working in compositions of abstract automata but also for all researchers interested in cryptosystems. Therefore, automata theoretical background of our results is not emphasized. The introduced cryptosystem is important also from a theoretical point of view, because it is the first fully functioning block cipher based on automata network.

Graffiti als Version und Subversion–Praxen kultureller Re-Regulierung und die Möglichkeit von Gratiforschung

2010 ◽  
Vol 55 (2) ◽  
pp. 11-25 ◽  
Author(s):  
Bernd Dollinger
Keyword(s):  
Social Life ◽  
Point Of View ◽  
Theoretical Point ◽  
Research Practice ◽  
Wird Eine ◽  
Multiple Perspectives

Der Beitrag geht von Versuchen aus, integrative Perspektiven einer überaus heterogenen Graffitiforschung zu bestimmen. In Auseinandersetzung insbesondere mit Bruno Latours Ansatz des »Iconoclash« wird eine kulturtheoretische Referenz bestimmt, die Graffiti als Version identifiziert, d. h. als semiotisch orientierte Veränderung räumlich situierter Ordnungs- und Regulierungspraxen. Ihnen kann, wenn auch nicht zwingend, eine subversive Qualität zukommen. Durch die Ausrichtung am Konzept einer Version wird beansprucht, Forderungen einer normativ weitgehend abstinenten, nicht-essentialistischen und für komplexe Fragen der Identitäts- und Raumpolitik offenen Forschungspraxis einzulösen.<br><br>The contribution attempts to integrate multiple perspectives of current largely heterogeneous graffiti scholarship. Referring to Bruno Latour’s concept »iconoclash«, we discuss graffiti from a cultural-theoretical point of view as a »version«. It appears as a semiotically oriented modification of spatially situated practices that regulate social life. Often, but not necessarily, these practices involve subversive qualities. The concept of »version« facilitates a non-normative and non-essentialist strategy of research. This enables an explorative research practice in which the complex matters of identity and space politics that are associated with graffiti can be addressed.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

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