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

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 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.

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].

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

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.

On the compressed essential graph of a module over a commutative ring

2020 ◽  
pp. 2150125
Author(s):  
S. H. Payrovi ◽  
S. Babaei ◽  
E. Sengelen Sevim
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Essential Ideal

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module. The compressed essential graph of [Formula: see text], denoted by [Formula: see text] is a simple undirected graph associated to [Formula: see text] whose vertices are classes of torsion elements of [Formula: see text] and two distinct classes [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, we study diameter and girth of [Formula: see text] and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that [Formula: see text], whenever [Formula: see text] is Noetherian and [Formula: see text] is a finitely generated multiplication module with [Formula: see text].

scholarly journals On Almost Semiprime Submodules

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Farkhonde Farzalipour
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Basic Properties ◽  
Nonzero Identity

We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules.

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