On Radicals of Submodules of Finitely Generated Modules

1986 ◽  
Vol 29 (1) ◽  
pp. 37-39 ◽  
Author(s):  
Roy L. McCasland ◽  
Marion E. Moore
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Prime Submodules ◽  
Finitely Generated Modules ◽  
The Ideal

AbstractThe concept of the M-radical of a submodule B of an R-module A is discussed (R is a commutative ring with identity and A is a unitary fl-module). The M-radical of B is defined as the intersection of all prime submodules of A containing B. The main result of the paper is that if denotes the ideal radical of (B:A), then M-rad B = provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module, where for some

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 $1\leq t\leq n$‎.

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 GORENSTEIN DIMENSIONS MODULO A REGULAR ELEMENT

2015 ◽  
Vol 99 (2) ◽  
pp. 260-266 ◽  
Author(s):  
SHAHAB RAJABI ◽  
SIAMAK YASSEMI
Keyword(s):  
Commutative Ring ◽  
Regular Element ◽  
Finitely Generated ◽  
Homological Dimensions ◽  
Finitely Generated Modules

Let $R$ be a commutative ring. In this paper we study the behavior of Gorenstein homological dimensions of a homologically bounded $R$-complex under special base changes to the rings $R_{x}$ and $R/xR$, where $x$ is a regular element in $R$. Our main results refine some known formulae for the classical homological dimensions. In particular, we provide the Gorenstein counterpart of a criterion for projectivity of finitely generated modules, due to Vasconcelos.

On 2-Absorbing Submodules

2012 ◽  
Vol 19 (spec01) ◽  
pp. 913-920 ◽  
Author(s):  
Sh. Payrovi ◽  
S. Babaei
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Ordered Set ◽  
Ordered Sets ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Totally Ordered Set

In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:RM. Let N be a 2-absorbing submodule of M. It is shown that N:RM is a 2-absorbing ideal of R and either Ass R(M/N) is a totally ordered set or Ass R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass R(M/N) a totally ordered set, then N is 2-absorbing whenever N:RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.

scholarly journals Pseudo-Noetherian Rings

1976 ◽  
Vol 19 (1) ◽  
pp. 77-84 ◽  
Author(s):  
Kenneth P. McDowell
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Finitely Generated Modules

In the latter part of the 1950’s some interesting papers appeared (e.g. [2] and [10]) which examined the relationships occurring between the purely algebraic and homological aspects of the theory of finitely generated modules over Noetherian rings. Many of these relationships remain valid if one considers the much wider class of rings determined by the following definition.Definition. A commutative ring R is called pseudo-Noetherian if it satisfies the following two conditions.

scholarly journals Primary modules over commutative rings

2001 ◽  
Vol 43 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Patrick F. Smith
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Radical ◽  
Finitely Generated ◽  
One Dimensional ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Commutative Domain

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

On the Maximal Spectrum of Semiprimitive Multiplication Modules

2008 ◽  
Vol 51 (3) ◽  
pp. 439-447
Author(s):  
Karim Samei
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Discrete Space ◽  
Multiplication Module ◽  
Prime Submodules ◽  
Maximal Submodule ◽  
Isolated Points ◽  
Maximal Spectrum ◽  
One Point Compactification

AbstractAnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.

scholarly journals THE SPACE OF FINITELY GENERATED RINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 373-382 ◽  
Author(s):  
YVES CORNULIER
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Metrizable Space ◽  
Finitely Generated ◽  
Compact Metrizable Space ◽  
Finitely Generated Modules

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor–Bendixson rank of any member of this space. For instance, the Cantor–Bendixson rank of the free commutative ring on n generators is ωn, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

scholarly journals The asymptotic closure of an ideal relative to a module

2001 ◽  
Vol 32 (3) ◽  
pp. 231-235
Author(s):  
Sylvia M. Foster ◽  
Johnny A. Johnson
Keyword(s):  
Commutative Ring ◽  
Integral Closure ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
The Ideal

In this paper we introduce the concept of the asymptotic closure of an ideal of a commutative ring $ R $ with identity relative to a unitary $ R $-module $ M $. This work extends results from P. Samuel, M. Nagata, J. W. Petro and Sharp, Tiras, and Yassi. Our objectives in this paper are to establish the cancellation law for the asymptotic completion of an ideal relative to a finitely generated module and show that the integral closure of an ideal relative to a Noetherian module $ M $ coincides with the asymptotic closure of the ideal relative to the Noetherian module $ M $.

Sign in / Sign up

Export Citation Format

Share Document