scholarly journals Half-exact coherent functors over Dedekind domains

2019 ◽  
Vol 18 (05) ◽  
pp. 1950099
Author(s):  
Adson Banda
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Dedekind Domain ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Coherent Functor ◽  
Coherent Functors

Let [Formula: see text] be a principal ideal domain (PID) or more generally a Dedekind domain and let [Formula: see text] be a coherent functor from the category of finitely generated [Formula: see text]-modules to itself. We classify the half-exact coherent functors [Formula: see text]. In particular, we show that if [Formula: see text] is a half-exact coherent functor over a Dedekind domain [Formula: see text], then [Formula: see text] is a direct sum of functors of the form [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is a finitely generated projective [Formula: see text]-module, [Formula: see text] a nonzero prime ideal in [Formula: see text] and [Formula: see text].

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

A CHARACTERIZATION OF GOLDIE EXTENDING MODULES OVER DEDEKIND DOMAINS

2011 ◽  
Vol 10 (06) ◽  
pp. 1291-1299 ◽  
Author(s):  
EVRIM AKALAN ◽  
GARY F. BIRKENMEIER ◽  
ADNAN TERCAN
Keyword(s):  
Principal Ideal ◽  
Dedekind Domain ◽  
Principal Ideal Domain ◽  
Direct Sums ◽  
Ideal Domain ◽  
Dedekind Domains ◽  
Pure Submodules ◽  
Torsion Modules

In this paper, we characterize [Formula: see text]-extending (Goldie extending) modules over Dedekind domains and we use the [Formula: see text]-extending condition to characterize the modules over a principal ideal domain whose pure submodules are direct summands. Moreover, we show that if R is a principal ideal domain, then the class of [Formula: see text]-extending modules is closed under direct summands and that if R is a Dedekind domain, then the class of [Formula: see text]-extending torsion modules is closed under finite direct sums.

Noetherian Rings in Which Every Ideal is a Product of Primary Ideals

1980 ◽  
Vol 23 (4) ◽  
pp. 457-459 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Primary Ideals

The classical rings of number theory, Dedekind domains, are characterized by the property that every ideal is a product of prime ideals. More generally, a commutative ring R with identity has the property that every ideal is a product of prime ideals if and only if R is a finite direct sum of Dedekind domains and special principal ideal rings. These rings, called general Z.P.I. rings, are also characterized by the property that every (prime) ideal is finitely generated and locally principal.

Strongly D2 modules and their applications

2021 ◽  
pp. 2250071
Author(s):  
Mingzhao Chen ◽  
Hwankoo Kim ◽  
Fanggui Wang
Keyword(s):  
Positive Integer ◽  
Principal Ideal ◽  
Projective Module ◽  
Negative Answer ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Finitely Generated Modules ◽  
Structural Characterizations ◽  
Semihereditary Rings

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

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 Graded 2-Prime Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 493
Author(s):  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Principal Ideal Domain ◽  
Graded Rings ◽  
Graded Ring ◽  
Ideal Domain ◽  
Primary Ideals ◽  
Weakly Prime Ideals

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over Z-graded principal ideal domain.

scholarly journals A note on almost prime submodule of CSM module over principal ideal domain

2021 ◽  
Vol 2106 (1) ◽  
pp. 012011
Author(s):  
I G A W Wardhana ◽  
N D H Nghiem ◽  
N W Switrayni ◽  
Q Aini
Keyword(s):  
Prime Ideal ◽  
Algebraic Structure ◽  
Principal Ideal ◽  
Ring Theory ◽  
Principal Ideal Domain ◽  
Module Theory ◽  
Ideal Domain ◽  
Multiplication Operation ◽  
Prime Submodule ◽  
Almost Prime

Abstract An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

Two Notes on Automorphisms of Modules over a PID

2019 ◽  
Vol 26 (03) ◽  
pp. 401-410
Author(s):  
Heguo Liu ◽  
Xiaoliang Luo ◽  
Xin Qin ◽  
Bomin Zan
Keyword(s):  
Principal Ideal ◽  
Minimal Condition ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Cyclic Submodules

Let D be a principal ideal domain (PID) and M be a module over D. We prove the following two dual results: (i) If M is finitely generated and x, y are two elements in M such that [Formula: see text], then there exists an automorphism α of M such that [Formula: see text]. (ii) If M satisfies the minimal condition on submodules and X, Y are two locally cyclic submodules of M such that [Formula: see text] and [Formula: see text], then there exists an automorphism α of M such that [Formula: see text].

scholarly journals Generalized equivalence of matrices over Prüfer domains

1991 ◽  
Vol 14 (4) ◽  
pp. 665-673 ◽  
Author(s):  
Frank DeMeyer ◽  
Hainya Kakakhail
Keyword(s):  
Commutative Ring ◽  
Equivalence Relation ◽  
Sufficient Conditions ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Ideal Domain ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Necessary And Sufficient ◽  
Invertible Matrices

Twom×nmatricesA,Bover a commutative ringRare equivalent in case there are invertible matricesP,QoverRwithB=PAQ. While anym×nmatrix over a principle ideal domain can be diagonalized, the same is not true for Dedekind domains. The first author and T. J. Ford introduced a coarser equivalence relation on matrices called homotopy and showed anym×nmatrix over a Dedekind domain is homotopic to a direct sum of1×2matrices. In this article give, necessary and sufficient conditions on a Prüfer domain that anym×nmatrix be homotopic to a direct sum of1×2matrices.

scholarly journals N Sequence Prime Ideals

2021 ◽  
pp. 3672-3678
Author(s):  
Hemin A. Ahmad ◽  
Parween A. Hummadi
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Closed Ideal ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Strongly Irreducible

    In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

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