Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

Ordinary Singularities with Decreasing Hilbert Function

1982 ◽  
Vol 34 (1) ◽  
pp. 169-180 ◽  
Author(s):  
Leslie G. Roberts
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Hilbert Function ◽  
Integral Closure ◽  
Minimal Prime Ideal ◽  
Quotient Field ◽  
Graded Ring ◽  
Image Position ◽  
Minimal Prime ◽  
Dimension One

Let A be the co-ordinate ring of a reduced curve over a field k. This means that A is an algebra of finite type over k, A has no nilpotent elements, and that if P is a minimal prime ideal of A, then A/P is an integral domain of Krull dimension one. Let M be a maximal ideal of A. Then G(A) (the graded ring of A relative to M) is defined to be . We get the same graded ring if we first localize at M, and then form the graded ring of AM relative to the maximal ideal MAM. That isLet Ā be the integral closure of A. If P1, P2, …, Ps are the minimal primes of A thenwhere A/Pi is a domain and is the integral closure of A/Pi in its quotient field.

A Theorem on Pure Submodules

1960 ◽  
Vol 12 ◽  
pp. 483-487
Author(s):  
George Kolettis
Keyword(s):  
Free Abelian Group ◽  
Chain Condition ◽  
Affirmative Answer ◽  
Principal Ideal Ring ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Pure Submodules ◽  
Torsion Free ◽  
Countable Case

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

scholarly journals Note on U-closed semigroup rings

1999 ◽  
Vol 59 (3) ◽  
pp. 467-471 ◽  
Author(s):  
Ryûki Matsuda
Keyword(s):  
Abelian Group ◽  
Integral Domain ◽  
Free Abelian Group ◽  
Semigroup Ring ◽  
Closed Domain ◽  
Semigroup Rings ◽  
Quotient Field ◽  
Torsion Free Abelian Group ◽  
Torsion Free

Let D be an integral domain with quotient field K. If α2 − α ∈ D and α3 − α2 ∈ D imply α ∈ D for all elements α of K, then D is called a u-closed domain. A submonoid S of a torsion-free Abelian group is called a grading monoid. We consider the semigroup ring D[S] of a grading monoid S over a domain D. The main aim of this note is to determine conditions for D[S] to be u-closed. We shall show the following Theorem: D[S] is u-closed if and only if D is u-closed.

scholarly journals A note on universally zero-divisor rings

1991 ◽  
Vol 43 (2) ◽  
pp. 233-239 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Integral Extension ◽  
Maximal Ideals ◽  
Unitary Module ◽  
Finite Set ◽  
Descending Chain Condition

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

scholarly journals On Schur's conjecture

1995 ◽  
Vol 58 (3) ◽  
pp. 312-357 ◽  
Author(s):  
Gerhard Turnwald
Keyword(s):  
Integral Domain ◽  
Elementary Proof ◽  
Prime Ideals ◽  
Quotient Field ◽  
Prime Degree ◽  
Dickson Polynomials ◽  
Quantitative Version ◽  
Linear Polynomials

AbstractWe study polynomials over an integral domainRwhich, for infinitely many prime idealsP, induce a permutation ofR/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field ofR. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

scholarly journals Noetherian and Artinian ordered groupoids—semigroups

2005 ◽  
Vol 2005 (13) ◽  
pp. 2041-2051
Author(s):  
Niovi Kehayopulu ◽  
Michael Tsingelis
Keyword(s):  
Chain Condition ◽  
Growth Conditions ◽  
Ideal Theory ◽  
Minimum Condition ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Maximal Ideals ◽  
Ordered Groupoid ◽  
Descending Chain Condition

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (semigroups) which are about to be introduced are particularly versatile. These satisfy a certain finiteness condition, namely, that every ideal of the ordered groupoid (semigroup) is finitely generated. Our purpose is to introduce the concepts of Noetherian and Artinian ordered groupoids. An ordered groupoid is said to be Noetherian if every ideal of it is finitely generated. In this paper, we prove that an equivalent formulation of the Noetherian requirement is that the ideals of the ordered groupoid satisfy the so-called ascending chain condition. From this idea, we are led in a natural way to consider a number of results relevant to ordered groupoids with descending chain condition for ideals. We moreover prove that an ordered groupoid is Noetherian if and only if it satisfies the maximum condition for ideals and it is Artinian if and only if it satisfies the minimum condition for ideals. In addition, we prove that there is a homomorphismπof an ordered groupoid (semigroup)Shaving an idealIonto the Rees quotient ordered groupoid (semigroup)S/I. As a consequence, ifSis an ordered groupoid andIan ideal ofSsuch that bothIand the quotient groupoidS/Iare Noetherian (Artinian), then so isS. Finally, we give conditions under which the proper prime ideals of commutative Artinian ordered semigroups are maximal ideals.

scholarly journals Krull modules and completely integrally closed modules

2021 ◽  
Author(s):  
Mu’amar Musa Nurwigantara ◽  
Indah Emilia Wijayanti ◽  
Hidetoshi Marubayashi ◽  
Sri Wahyuni
Keyword(s):  
Integral Domain ◽  
Free Module ◽  
Minimal Prime Ideal ◽  
Closed Formula ◽  
Multiplication Module ◽  
Quotient Field ◽  
Integrally Closed ◽  
Torsion Free Module ◽  
Minimal Prime ◽  
Torsion Free

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

Primitive Near-rings — Some Structure Theorems

2007 ◽  
Vol 14 (03) ◽  
pp. 417-424 ◽  
Author(s):  
Gerhard Wendt
Keyword(s):  
Structure Theorem ◽  
General Structure ◽  
Chain Condition ◽  
Multiplicative Semigroup ◽  
Function Composition ◽  
Regular Elements ◽  
Structure Theorems ◽  
Descending Chain Condition ◽  
A Chain

We show that any zero symmetric 1-primitive near-ring with descending chain condition on left ideals can be described as a centralizer near-ring in which the multiplication is not the function composition but sandwich multiplication. This result follows from a more general structure theorem on 1-primitive near-rings with multiplicative right identity, not necessarily having a chain condition on left ideals. We then use our results to investigate more closely the multiplicative semigroup of a 1-primitive near-ring. In particular, we show that the set of regular elements forms a right ideal of the multiplicative semigroup of the near-ring.

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