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.

scholarly journals Nice Bases of QTAG-Modules

2021 ◽  
pp. 1-5
Author(s):  
Fahad Sikander ◽  
Tanveer Fatima ◽  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Divisible Hull

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

scholarly journals On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Firdhousi Begum
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Pure Submodules

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

Structures of QTAG-modules determined by their submodules

2021 ◽  
Vol 65 (3) ◽  
pp. 38-45
Author(s):  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

scholarly journals A Note on Elongations of SummableQTAG-Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Sabah A. R. K. Naji
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Suitable Condition ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules

A right moduleMover an associative ring with unity is aQTAG-module if every finitely generated submodule of any homomorphic image ofMis a direct sum of uniserial modules. In this paper we find a suitable condition under which a specialω-elongation of a summableQTAG-module by aω+k-projectiveQTAG-module is also a summableQTAG-module.

scholarly journals Different Characterizations of Large Submodules of QTAG-Modules

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Fahad Sikander ◽  
Alveera Mehdi ◽  
Sabah A. R. K. Naji
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Necessary And Sufficient

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule L of M is large in M if L+B=M, for every basic submodule B of M. The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on.

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

scholarly journals Projective summands in generators

1982 ◽  
Vol 86 ◽  
pp. 203-209 ◽  
Author(s):  
David Eisenbud ◽  
Wolmer Vasconcelos ◽  
Roger Wiegand
Keyword(s):  
Homomorphic Image ◽  
Polynomial Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Positive Direction ◽  
Chain Conditions ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Category Of Modules

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.

scholarly journals Symmetries of Kirchberg Algebras

2003 ◽  
Vol 46 (4) ◽  
pp. 509-528 ◽  
Author(s):  
David J. Benson ◽  
Alex Kumjian ◽  
N. Christopher Phillips
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Direct Sum ◽  
Representation Ring ◽  
Direct Sum Decomposition ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
The Way

AbstractLet G0 and G1 be countable abelian groups. Let γi be an automorphism of Gi of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1A] = 0 in K0(A), and an automorphism α ∈ Aut(A) of order two, such that K0(A) ≅ G0, such that K1(A) ≅ G1, and such that α* : Ki(A) → Ki(A) is γi. As a consequence, we prove that every -graded countable module over the representation ring R() of is isomorphic to the equivariant K-theory K (A) for some action of on a unital Kirchberg algebra A.Along the way, we prove that every not necessarily finitely generated []-module which is free as a -module has a direct sum decomposition with only three kinds of summands, namely [] itself and on which the nontrivial element of acts either trivially or by multiplication by −1.

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.

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350057
Author(s):  
SURJEET SINGH
Keyword(s):  
Homomorphic Image ◽  
Artinian Ring ◽  
Finitely Generated ◽  
Serial Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Universal Construction ◽  
Generating System

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

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