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).

scholarly journals Injective Modules Over Non-Artinian Serial Rings

1988 ◽  
Vol 44 (2) ◽  
pp. 242-251 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Affirmative Answer ◽  
Primitive Idempotent ◽  
Injective Hull ◽  
Finitely Generated ◽  
Serial Ring ◽  
Direct Sum ◽  
Injective Modules

AbstractA module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

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).

A Note on Compactifying Artinian Rings

1974 ◽  
Vol 26 (3) ◽  
pp. 580-582 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Finite Group ◽  
Artinian Ring ◽  
Topological Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
A Finite Group

In this note a number of compactifications are discussed within the class of artinian rings. In [1] the following was proved:Theorem. For an artinian ring R the following are equivalent:(1) R is equationally compact.(2) R+ ≃ B ⊕ P, where B is a finite group, P is a finite direct sum of Prüfer groups, and R · P = P · R = {0}.(3) R is a retract of a compact topological ring.

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 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.

scholarly journals Non-local rings whose ideals are all quasi-injective

1972 ◽  
Vol 6 (1) ◽  
pp. 45-52 ◽  
Author(s):  
G. Ivanov
Keyword(s):  
Finite Number ◽  
Local Rings ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Left And Right

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

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 Some characterizations of hereditarily artinian rings

1986 ◽  
Vol 28 (1) ◽  
pp. 21-23 ◽  
Author(s):  
Dinh van Huynh
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Descending Chain Condition ◽  
Associative Rings

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

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