scholarly journals Integral closures of ideals relative to Artinian modules, and exact sequences

1992 ◽  
Vol 34 (1) ◽  
pp. 103-107 ◽  
Author(s):  
R. Y. Sharp ◽  
Y. Tiraş
Keyword(s):  
Commutative Ring ◽  
Integral Closure ◽  
Artinian Modules ◽  
Integral Closures ◽  
Exact Sequences

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

A theorem on integral closures of ideals in semi-local rings

1979 ◽  
Vol 85 (1) ◽  
pp. 55-59 ◽  
Author(s):  
A. Lindsay Burch
Keyword(s):  
Commutative Ring ◽  
Integral Closure ◽  
Local Rings ◽  
Integral Closures

Consider an ideal A of a commutative ring R; denote by Ai the integral closure of A in R.

scholarly journals A note on reductions of ideals relative to an Artinian module

1993 ◽  
Vol 35 (2) ◽  
pp. 219-224 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Integral Closure ◽  
Artinian Modules ◽  
Artinian Module ◽  
Image Position ◽  
Positive Integers

The concept of reduction and integral closure of ideals relative to Artinian modules were introduced in [7]; and we summarize some of the main aspects now.Let A be a commutative ring (with non-zero identity) and let a, b be ideals of A. Suppose that M is an Artinian module over A. We say that a is a reduction of b relative to M if a ⊆ b and there is a positive integer s such that)O:Mabs)=(O:Mbs+l).An element x of A is said to be integrally dependent on a relative to M if there exists n y ℕ(where ℕ denotes the set of positive integers) such thatIt is shown that this is the case if and only if a is a reduction of a+Ax relative to M; moreoverᾱ={x ɛ A: xis integrally dependent on a relative to M}is an ideal of A called the integral closure of a relative to M and is the unique maximal member of℘ = {b: b is an ideal of A which has a as a reduction relative to M}.

Asymptotic Behavior of Integral Closures in Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 505-514 ◽  
Author(s):  
R. Naghipour ◽  
P. Schenzel
Keyword(s):  
Asymptotic Behavior ◽  
Integral Closure ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Large N ◽  
Nagata Ring ◽  
Associated Prime Ideals ◽  
Integral Closures ◽  
Definition Of ◽  
Associated Prime

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

Artinian modules over commutative rings

1992 ◽  
Vol 111 (1) ◽  
pp. 25-33 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Matlis Duality ◽  
Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

scholarly journals BILINEAR MAPS ON ARTINIAN MODULES

2012 ◽  
Vol 11 (05) ◽  
pp. 1250090
Author(s):  
GEORGE M. BERGMAN
Keyword(s):  
Commutative Ring ◽  
Finite Length ◽  
Nonzero Element ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Noncommutative Rings ◽  
Artinian Modules

It is shown that if a bilinear map f : A × B → C of modules over a commutative ring k is nondegenerate (i.e. if no nonzero element of A annihilates all of B, and vice versa), and A and B are Artinian, then A and B are of finite length. Some consequences are noted. Counterexamples are given to some attempts to generalize the above result to balanced bilinear maps of bimodules over noncommutative rings, while the question is raised whether other such generalizations are true.

The commutative ring of a finite projective plane

1985 ◽  
Vol 101 (1-2) ◽  
pp. 57-59 ◽  
Author(s):  
Alan R. Prince
Keyword(s):  
Projective Plane ◽  
Commutative Ring ◽  
Finite Projective Plane ◽  
Integral Closure ◽  
Gorenstein Ring

SynopsisThe ring of a finite projective plane, introduced by the author in a previous paper, is shown to be a Gorenstein ring. Its integral closure and conductor are also determined.

ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS

2002 ◽  
Vol 01 (04) ◽  
pp. 451-467 ◽  
Author(s):  
PAOLO ZANARDO
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Closure ◽  
Nonzero Ideal ◽  
Valuation Domain ◽  
Local Domain ◽  
Valuation Domains ◽  
Domain V

A commutative ring R is said to be almost perfect if R/I is perfect for every nonzero ideal I of R. We prove that an almost perfect local domain R is dominated by a unique archimedean valuation domain V of its field of quotients Q if and only if the integral closure of R contains an ideal of V. We show how to construct almost perfect local domains dominated by finitely many archimedean valuation domains. We provide several examples illustrating various possible situations. In particular, we construct an almost perfect local domain whose maximal ideal is not almost nilpotent.

scholarly journals Key Exchange Over Particular Algebraic Closure Ring

2017 ◽  
Vol 70 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Mohammed Sahmoudi ◽  
Abdelhakim Chillali
Keyword(s):  
Commutative Ring ◽  
Algebraic Closure ◽  
Integral Closure ◽  
Key Exchange ◽  
New Method ◽  
Secret Key ◽  
Diffie Hellman ◽  
Diffie Hellman Key Exchange

Abstract In this paper, we propose a new method of Diffie-Hellman key exchange based on a non-commutative integral closure ring. The key idea of our proposal is that for a given non-commutative ring, we can define the secret key and take it as a common key to encrypt and decrypt the transmitted messages. By doing, we define a new non-commutative structure over the integral closure OL of sextic extension L, namely L is an extension of ℚ of degree 6 in the form ℚ(α, β), which is a rational quadratic and monogenic extension over a non-pure and monogenic cubic subfield K = ℚ(β).

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