scholarly journals Krull's principal ideal theorem in non-Noetherian settings

2018 ◽  
Vol 168 (1) ◽  
pp. 13-27
Author(s):  
BRUCE OLBERDING
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated

AbstractLetPbe a finitely generated ideal of a commutative ringR. Krull's principal ideal theorem states that ifRis Noetherian andPis minimal over a principal ideal ofR, thenPhas height at most one. Straightforward examples show that this assertion fails ifRis not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules

2020 ◽  
Vol 27 (1) ◽  
pp. 103-110
Author(s):  
Shahram Motmaen ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Module Homomorphism ◽  
Prime Submodule ◽  
Bezout Module

AbstractIn this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set\mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}.For an R-module {M\in\mathbb{H}}, set {T=(R\setminus Z(R))\cap(R\setminus Z(M))}, {\mathfrak{T}(M)=T^{-1}M} and {P=(\mathrm{Nil}(M):_{R}M)}. In this case, the mapping {\Phi:\mathfrak{T}(M)\to M_{P}} given by {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into {M_{P}} given by {\Phi(x)=x/1} for every {x\in M}. A nonnil submodule N of M is said to be Φ-invertible if {\Phi(N)} is an invertible submodule of {\Phi(M)}. Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if {\Phi(N)} is a principal ideal of {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.

scholarly journals On some properties of⊕-supplemented modules

2003 ◽  
Vol 2003 (69) ◽  
pp. 4373-4387 ◽  
Author(s):  
A. Idelhadj ◽  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Principal Ideal Ring ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Ideal Ring

A moduleMis⊕-supplemented if every submodule ofMhas a supplement which is a direct summand ofM. In this paper, we show that a quotient of a⊕-supplemented module is not in general⊕-supplemented. We prove that over a commutative ringR, every finitely generated⊕-supplementedR-moduleMhaving dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ringRis semisimple if and only if the class of⊕-supplementedR-modules coincides with the class of injectiveR-modules. The structure of⊕-supplemented modules over a commutative principal ideal ring is completely determined.

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.

scholarly journals Torsion and protorsion modules over free ideal rings

1970 ◽  
Vol 11 (4) ◽  
pp. 490-498
Author(s):  
P. M. Cohn
Keyword(s):  
Principal Ideal ◽  
Finitely Generated ◽  
Commutative Case ◽  
Principal Ideal Domains ◽  
Torsion Modules ◽  
Noncommutative Analogue

Free ideal rings (or firs, cf. [2, 3] and § 2 below) form a noncommutative analogue of principal ideal domains, to which they reduce in the commutative case, and in [3] a category TR of right R-modules was defined, over any fir R, which forms an analogue of finitely generated torsion modules. The category TR was shown to be abelian, and all its objects have finite composition length; more over, the corresponding category RT of left R-modules is dual to TR.

Primary ideals with finitely generated radical in a commutative ring

1993 ◽  
Vol 78 (1) ◽  
pp. 201-221 ◽  
Author(s):  
Robert Gilmer ◽  
William Heinzer
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Primary Ideals

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

Invariant Factors Under Rank One Perturbations

1980 ◽  
Vol 32 (1) ◽  
pp. 240-245 ◽  
Author(s):  
Robert C. Thompson
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Matrix Multiplication ◽  
Diagonal Form ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Zero Divisors ◽  
Rank One ◽  
Invariant Factors

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

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