On commutative rings whose ideals are direct sums of cyclic modules

2019 ◽  
Vol 18 (02) ◽  
pp. 1950039
Author(s):  
A. Ghorbani ◽  
M. Naji-Esfahani
Keyword(s):  
Maximal Ideal ◽  
Structure Theorem ◽  
Commutative Rings ◽  
Local Dimension ◽  
Direct Sums ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Dimension Formula ◽  
Local Case ◽  
The Family

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension [Formula: see text]. Then we conclude that, for a ring of local dimension [Formula: see text], every ideal of [Formula: see text] is a direct sum of cyclic modules if and only if every indecomposable ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if [Formula: see text] is a principal ideal ring.

Commutative rings whose proper ideals are direct sum of completely cyclic modules

2016 ◽  
Vol 15 (09) ◽  
pp. 1650160 ◽  
Author(s):  
M. Behboodi ◽  
S. Heidari ◽  
S. Roointan-Isfahani
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Simple Formula ◽  
Commutative Rings ◽  
Index Set ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring

By two results of Köthe and Cohen–Kaplansky we obtain that “a commutative ring [Formula: see text] has the property that every [Formula: see text]-module is a direct sum of (completely) cyclic modules if and only if [Formula: see text] is an Artinian principal ideal ring” (an [Formula: see text]-module [Formula: see text] is called completely cyclic if each submodule of [Formula: see text] is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring [Formula: see text] is a direct sum of completely cyclic [Formula: see text]-modules if and only if [Formula: see text] is a principal ideal ring or [Formula: see text] is a local ring with maximal ideal [Formula: see text] such that there is an index set [Formula: see text] and a set of elements [Formula: see text] such that [Formula: see text] with each [Formula: see text] a simple [Formula: see text]-module and [Formula: see text] a principal ideal ring.

Direct Sums of Torsion-Free Covers

1973 ◽  
Vol 25 (5) ◽  
pp. 1002-1005
Author(s):  
Thomas Cheatham
Keyword(s):  
Direct Product ◽  
Commutative Rings ◽  
Integral Domains ◽  
Direct Sums ◽  
Artinian Rings ◽  
Direct Sum ◽  
Special Case ◽  
Torsion Free

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

On noncommutative FGC rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550109 ◽  
Author(s):  
A. Ghorbani ◽  
M. Naji Esfahani
Keyword(s):  
Open Problem ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Direct Sums ◽  
Principal Ideal Ring ◽  
Uniserial Ring ◽  
Ideal Ring ◽  
Special Cases ◽  
Finitely Generated Modules

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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.

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?”.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

scholarly journals Reduction to Dimension Two of the Local Spectrum for an AH Algebra with the Ideal Property

2017 ◽  
Vol 60 (4) ◽  
pp. 791-806 ◽  
Author(s):  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Local Spectrum ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Ideal Property ◽  
Uniformly Bounded ◽  
The Ideal

AbstractA C*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit C*-algebra A of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that A can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion H2 groups.

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