scholarly journals Notes on weakly-semisimple rings

1995 ◽  
Vol 52 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Yiqiang Zhou
Keyword(s):  
Noetherian Ring ◽  
Cyclic Module ◽  
Finitely Generated

Responding to a question on right weakly semisimple rings due to Jain, Lopez-Permouth and Singh, we report the existence of a non-right-Noetherian ring R for which every uniform cyclic right it-module is weakly-injective and every uniform finitely generated right R-module is compressible. We show that a ring R is a right Noetherian ring for which every cyclic right R-module is weakly R-injective if and only if R is a right Noetherian ring for which every uniform cyclic right R-module is compressible if and only if every cyclic right R-module is compressible. Finally, we characterise those modules M for which every finitely generated (respectively, cyclic) module in σ[M] is compressible.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

Artinianness of Composed Graded Local Cohomology Modules

2016 ◽  
Vol 59 (2) ◽  
pp. 271-278
Author(s):  
Fatemeh Dehghani-Zadeh
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Local Cohomology Modules ◽  
Base Ring ◽  
Cohomology Modules ◽  
Local Base

AbstractLet be a graded Noetherian ring with local base ring (R0 ,m0) and let . Let M and N be finitely generated graded R-modules and let a = a0 + R+ an ideal of R. We show that and are Artinian for some i s and j s with a specified property, where bo is an ideal of R0 such that a0 + b0 is an m0-primary ideal.

Ideals with Trivial Conormal Bundle

1980 ◽  
Vol 32 (1) ◽  
pp. 210-218 ◽  
Author(s):  
A. V. Geramita ◽  
C. A. Weibel
Keyword(s):  
Polynomial Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Question Answering ◽  
General Theorem ◽  
Closed Field ◽  
Natural Question ◽  
Finitely Generated ◽  
Algebraically Closed Field ◽  
Conormal Bundle

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

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].

scholarly journals On the length of the powers of systems of parameters in local ring

1990 ◽  
Vol 120 ◽  
pp. 77-88 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Finitely Generated ◽  
Systems Of Parameters ◽  
System Of Parameters ◽  
The Ideal

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A-module with dim (M) = d. Let x1, …, xd be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x1, …, xd.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

On ideals preserving generalized local cohomology modules

2015 ◽  
Vol 15 (01) ◽  
pp. 1650019 ◽  
Author(s):  
Tsutomu Nakamura
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ωt the set of ideals 𝔠 such that [Formula: see text] for all i < t. First, we show that there exists the ideal 𝔟t which is the largest in Ωt and [Formula: see text]. Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟t, then [Formula: see text] for all i < t.

On the Finiteness Dimension of Local Cohomology Modules

2014 ◽  
Vol 21 (03) ◽  
pp. 517-520 ◽  
Author(s):  
Hero Saremi ◽  
Amir Mafi
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Negative Integer ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.

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