AN AFFIRMATIVE ANSWER TO A QUESTION ON NOETHERIAN RINGS

2008 ◽  
Vol 07 (01) ◽  
pp. 47-59 ◽  
Author(s):  
DINH VAN HUYNH ◽  
S. TARIQ RIZVI
Keyword(s):  
Projective Module ◽  
Affirmative Answer ◽  
Noetherian Rings ◽  
Direct Sum

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is a direct sum of a projective module and a module Q, where Q is either injective or noetherian. This provides an affirmative answer to a question raised by P. F. Smith.

scholarly journals A characterization of noetherian rings by cyclic modules

1996 ◽  
Vol 39 (2) ◽  
pp. 253-262 ◽  
Author(s):  
Dinh Van Huynh
Keyword(s):  
Projective Module ◽  
Noetherian Rings ◽  
Noetherian Module ◽  
Direct Sum

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

scholarly journals Characterizing rings by a direct decomposition property of their modules

2006 ◽  
Vol 80 (3) ◽  
pp. 359-366 ◽  
Author(s):  
Dinh Van Huynh ◽  
S. Tariq Rizvi
Keyword(s):  
Projective Module ◽  
Direct Decomposition ◽  
Decomposition Property ◽  
Direct Sum ◽  
Continuous Module

AbstractA module M is said to satisfy the condition (℘*) if M is a direct sum of a projective module and a quasi-continuous module. In an earlier paper, we described the structure of rings over which every (countably generated) right module satisfies (℘*), and it was shown that such a ring is right artinian. In this note some additional properties of these rings are obtained. Among other results, we show that a ring over which all right modules satisfy (℘*) is also left artinian, but the property (℘*) is not left-right symmetric.

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.

A Theorem on Pure Submodules

1960 ◽  
Vol 12 ◽  
pp. 483-487
Author(s):  
George Kolettis
Keyword(s):  
Free Abelian Group ◽  
Chain Condition ◽  
Affirmative Answer ◽  
Principal Ideal Ring ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Pure Submodules ◽  
Torsion Free ◽  
Countable Case

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

scholarly journals When every projective module is a direct sum of finitely generated modules

2007 ◽  
Vol 315 (1) ◽  
pp. 454-481 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Gena Puninski ◽  
Philipp Rothmaler
Keyword(s):  
Projective Module ◽  
Finitely Generated ◽  
Direct Sum ◽  
Finitely Generated Modules

scholarly journals RINGS OVER WHICH CYCLICS ARE DIRECT SUMS OF PROJECTIVE AND CS OR NOETHERIAN

2010 ◽  
Vol 52 (A) ◽  
pp. 103-110 ◽  
Author(s):  
C. J. HOLSTON ◽  
S. K. JAIN ◽  
A. LEROY
Keyword(s):  
Projective Module ◽  
Regular Ring ◽  
Artinian Ring ◽  
Von Neumann Regular Ring ◽  
Finitely Generated Module ◽  
Direct Sums ◽  
Von Neumann ◽  
Noetherian Module ◽  
Direct Sum ◽  
Von Neumann Regular

AbstractR is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

scholarly journals Rings characterised by semiprimitive modules

1995 ◽  
Vol 52 (1) ◽  
pp. 107-116
Author(s):  
Yasuyuki Hirano ◽  
Dinh Van Huynh ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Module ◽  
Direct Sum

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

scholarly journals On the Ideal Case of a Conjecture of Huneke and Wiegand

2019 ◽  
Vol 62 (3) ◽  
pp. 847-859 ◽  
Author(s):  
Olgur Celikbas ◽  
Shiro Goto ◽  
Ryo Takahashi ◽  
Naoki Taniguchi
Keyword(s):  
Free Module ◽  
Affirmative Answer ◽  
Noetherian Rings ◽  
Ideal Case ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Integrally Closed ◽  
Integrally Closed Ideals ◽  
Torsion Free ◽  
The Ideal

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

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).

ON G-TYPE DOMAINS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250005 ◽  
Author(s):  
O. A. S. KARAMZADEH ◽  
B. MOSLEMI
Keyword(s):  
Prime Ideal ◽  
Noetherian Rings ◽  
Local Rings ◽  
Quotient Field ◽  
Basic Properties ◽  
Direct Sum ◽  
Noetherian Dimension ◽  
Hilbert Nullstellensatz

In this paper, we introduce and study the notion of G-type domains (a domain R is G-type if its quotient field is countably generated R-algebra). We extend some of the basic properties of G-domains to G-type domains. It's observed that a prime ideal of R[x1, x2,…,xn,…] is G-type if and only if its contractions in R, R[x1, x2,…,xn] for all n ≥ 1 are G-type. Using this concept we give a natural proof of the well-known Hilbert Nullstellensatz in infinite countable-dimensional spaces. Characterizations of Noetherian G-type domains, Noetherian G-type domains with the countable prime avoidance property are given. As a consequence, we observe that in complete Noetherian semi-local rings, G-type ideals and G-ideals are the same. Rings with countable Noetherian dimension which are direct sum of G-type domains are fully determined. Finally, we characterize Noetherian rings in which G-type ideals are maximal.

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